Associative polymers for use in a flow and related compositions, methods and systems

ABSTRACT

Described herein are associative polymers capable of controlling a physical and/or chemical property of non-polar compositions that can be used when the non-polar composition is in a flow, and related compositions, methods and systems. Associative polymers herein described have a non-polar backbone with a longest span having a molecular weight that remains substantially unchanged under the flow conditions and functional groups presented at ends of the non-polar backbone, with a number of the functional groups presented at the ends of the non-polar backbone formed by associative functional groups capable of undergoing an associative interaction with another associative functional group with an association constant (k) such that the strength of each associative interaction is less than the strength of a covalent bond between atoms and in particular less than the strength of a covalent bond between backbone atoms.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation application of U.S.application Ser. No. 16/526,806 filed on Jul. 30, 2019 which is acontinuation application of U.S. application Ser. No. 15/269,937 filedon Sep. 19, 2016, which issued as U.S. Pat. No. 10,428,286 on Oct. 1,2019, which claims priority to the U.S. Provisional Application Ser. No.62/236,099 entitled “Associative Polymers to Control Formation ofParticulate Matter from Ignitable Compositions and Related Compositions,Methods And Systems” filed on Oct. 1, 2015 with docket number P1173-USP4and the U.S. Provisional Application Ser. No. 62/220,922 entitled“Associative Polymers to and Related Compositions, Methods And Systems”filed on Sep. 18, 2015 with docket number P1173-USP3 and may be relatedto PCT Application No. PCT/US2016/052554 entitled “Associative Polymersfor Use in a Flow and Related Compositions Methods and Systems” filed onSep. 19, 2016 with Docket No. P1924-PCT, to U.S. application Ser. No.15/269,911 entitled “Associative Polymers To Control Formation OfParticulate Matter From Ignitable Compositions And Related Compositions,Methods And Systems” filed on Sep. 19, 2016 with Docket No. P1925-USwhich issued as U.S. Pat. No. 10,119,084 on Nov. 6, 2018, to PCTApplication No. PCT/US2016/052547 entitled “Associative Polymers ToControl Formation Of Particulate Matter From Ignitable Compositions AndRelated Compositions, Methods And Systems” filed on Sep. 19, 2016 withDocket No. P1925-PCT, to U.S. Non-Provisional application Ser. No.14/859,181 entitled “Associative Polymers and Related Compositions,Methods and Systems” filed on Sep. 18, 2015 with docket number P1759-US,to PCT International Application No. PCT/US15/51088 entitled“Associative Polymers and Related Compositions, Methods and Systems”filed on Sep. 18, 2015 with docket number P1759-PCT, and PCTInternational Application No. PCT/US15/51079 entitled “AssociativePolymers and Related Compositions, Methods and Systems” filed on Sep.18, 2015 with docket number P1760-PCT which claims priority to the U.S.Provisional Application Ser. No. 62/052,355 entitled “AssociativePolymers and Related Compositions, Methods and Systems” filed on Sep.18, 2014 with docket number P1173-USP2, which may be related toprovisional application 61/799,670 entitled “Associative Polymers andrelated Compositions Methods and Systems” filed on Mar. 15, 2013 withdocket number P1173-USP, to U.S. application Ser. No. 14/217,142entitled Associative Polymers and related Compositions Methods andSystems” filed on Mar. 17, 2014 with docket number P1173-US which issuedas U.S. Pat. No. 10,087,310 on Oct. 2, 2018, and to PCT application S/NPCT/US14/30772, entitled Associative Polymers and related CompositionsMethods and Systems” filed on Mar. 17, 2014 with docket numberP1173-PCT, the contents of each of which is incorporated herein byreference in its entirety.

STATEMENT OF GOVERNMENT GRANT

This invention was made with government support under Grant No.80NM0018D004 awarded by NASA (JPL). The government has certain rights inthe invention.

FIELD

The present disclosure relates to associative polymers for use in a flowand related compositions methods and systems. In particular, the presentdisclosure relates to associative polymers suitable to be used inconnection with control of physical and/or chemical properties ofnon-polar compositions.

BACKGROUND

Several non-polar compositions are known in the art for which control ofthe related physical and/or chemical properties is desired in particularwhen the non-polar composition is in a flow. For example, in hydrocarboncompositions which can be used for combustion and energy production,control of properties such as mist, drag, and combustion can bedesirable.

Also in non-polar liquid hydrocarbon compositions suitable to be used asink, pesticide or fuel, control of properties such as mist and dropbreakup can be desirable in particular when the the liquid hydrocarboncomposition is in a flow.

However, despite development of several approaches, control of thoseproperties in liquid composition in a flow is still challenging.

SUMMARY

Provided herein are associative polymers which in several embodimentscan be used as additives in a non-polar composition, and relatedcompositions, methods, and systems. In particular associative polymersherein described in several embodiments allows control of physicaland/or chemical properties, and in particular rheological properties,and are particularly effective when the non-polar composition is in aflow, thus allowing for example drag reduction, mist control,lubrication, fuel efficiency and/or control of viscoelastic propertiesof a non-polar composition.

In general associative polymers herein described have a non-polarbackbone and functional groups presented at ends of the non-polarbackbone, with a number of the functional groups presented at the endsof the non-polar backbone being associative functional groups. Anassociative functional group in associative polymers herein describedare capable of undergoing an associative interaction with anotherassociative functional group with an association constant (k) such thatthe strength of each associative interaction is less than the strengthof a covalent bond between atoms and in particular less than thestrength of a covalent bond between backbone atoms. In particular, inassociative polymers herein described associative functional groups canhave an association constant (k)

${k\left( M^{- 1} \right)} \geq {\frac{\frac{4}{3}{\pi\left( {R_{g}}^{2} \right)}^{\frac{3}{2}}N_{a}}{n_{F}} \times 10^{- 23}}$

in which R_(g) is the radius of gyration of the associative polymer in anon-polar composition (R_(g) in nanometers), N_(a) is Avogadro'sconstant; and n_(F) is the average number of the associative functionalgroups in the associative polymer. In some embodiments, an associativepolymer herein described can have an overall weight average molecularweight, M_(w), equal to or lower than about 2,000,000 g/mol, and/or aM_(w) equal to or higher than about 100,000 g/mol.

According to a first aspect, a linear or branched associative polymer isdescribed, herein also indicated as framing associative polymer, whichcomprises a linear, branched, or hyperbranched backbone having at leasttwo ends and functional groups presented at two or more ends of the atleast two ends of the backbone. In the framing associative polymer, thelinear or branched polymer backbone is substantially soluble in anon-polar composition, in particular in a host non polar composition,and a number of the functional groups presented at the two or more endsof the of the at least two ends of the backbone is formed by associativefunctional groups, wherein a longest span of the framing associativepolymer has a contour length L_(f), such that ½L_(bf)≤L_(f)<L_(bf),wherein L_(bf) is a rupture length of the framing associative polymer innanometers (nm) when the framing associative polymer is comprised withinthe host non-polar composition at framing associative polymerconcentration c to provide an associative non-polar composition in aflow, L_(bf) being given by implicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons (nN), Re is the Reynolds number, d is the characteristiclength of the flow in meters (m), μ is the viscosity of the hostnon-polar composition μ_(h) or the viscosity of the associative nonpolar composition μ_(a) in Pascal second (Pa·s), and ρ is the density ofthe host non-polar composition ρ_(h) or the viscosity of the associativenon polar composition Pa in Kilogram/meter³ (kg/m³).In associative polymers herein described, when c≤2c*, μ is the viscosityof the host non-polar composition μ_(h), and ρ is the density of thehost non-polar composition ρ_(h), and when c>2c*, μ is the viscosity ofthe associative non-polar composition μ_(a), and ρ is the density of theassociative non-polar composition μ_(a).In some embodiments, the linear or branched framing associative polymerhas an overall weight average molecular weight, M_(w), is equal to orlower than about 2,000,000 g/mol.

According to a second aspect, a linear or branched associative polymeris described, herein also indicated as capping associative polymer,which comprises a linear, branched, or hyperbranched polymer backbonehaving at least two ends and an associative functional group presentedat one end of the at least two ends of the backbone. In the cappingassociative polymer, the linear or branched backbone is substantiallysoluble in a non-polar composition and in particular in a host non polarcomposition. In some embodiments the capping associative polymer has anoverall weight-average molecular weight, M_(w), equal to or lower thanabout 2,000,000 g/mol, and/or M_(w) equal to or higher than about100,000 g/mol. In some embodiments, the terminal linear or branchedassociative polymer is a linear polymer. In some embodiments, a longestspan of the capping associative polymer has a contour length L_(c), suchthat ½L_(bc)≤L_(c)<L_(bc), wherein L_(bc) is a rupture length of thecapping associative polymer in nanometers, when the capping associativepolymer is comprised within the host non-polar composition together withat least one framing associative polymer at a framing associativepolymer concentration c to provide an associative non-polar compositionin a flow, L_(bc) being given by implicit function

$F_{bc} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bc} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{bc} \right)}} \times 10^{- 9}}$

in which F_(bc) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.In embodiments wherein a longest span of the capping associative polymerhas a contour length L_(c), such that ½L_(bc)≤L<L_(bc), when c≤2c*, μ isthe viscosity of the host non-polar composition μ_(h), ρ is the densityof the host non-polar composition ρ_(h), and when c>2c*, μ is theviscosity of the associative non-polar composition μ_(a), and ρ is thedensity of the associative non-polar composition ρ_(a).In some embodiments, the linear or branched framing associative polymerhas an overall weight average molecular weight, M_(w), equal to or lowerthan about 2,000,000 g/mol.

According to a third aspect, any one of the associative polymers hereindescribed and in particular any one of the framing associative polymersand/or capping associative polymers herein described, can have aweight-average molecular weight equal to or lower than 1,000,000 g/mol.In those embodiments, associative polymer herein described can be shearresistant depending on the structure of the backbone and on thepresence, number and location of secondary, tertiary and quaternarycarbon atoms in backbone. In some embodiments, framing associativepolymers and/or capping associative polymers herein described can have aweight-average molecular weight equal to or lower than 750,000 g/mol. Insome embodiments, framing associative polymers and/or cappingassociative polymers herein described can have a weight-averagemolecular weight between 400,000 g/mol and 1,000,000 g/mol.

According to a fourth aspect an associative (or modified) non-polarcomposition is described, the associative non-polar compositioncomprising a host composition having a viscosity μ_(h), a density ρ_(h),and a dielectric constant equal to or less than about 5 and at least oneframing associative polymer herein described, and optionally at leastone capping associative polymer herein described, the at least oneframing associative polymer and the at least one capping associativepolymer substantially soluble in the host composition. In particular, inthe associative non polar composition, the longest span of the at leastone framing associative polymer has a countour length½L_(bf)≤L_(f)<L_(bf), wherein L_(bf) is a rupture length of the framingassociative polymer in nanometers when the framing associative polymeris within the host non-polar composition at a concentration c to providethe associative non-polar composition in a flow, L_(bf) being given byimplicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.In the associative non-polar composition herein described, the at leastone framing associative polymer herein described can be comprised in thehost composition at a concentration from about 0.01 c* to 10c*, withrespect to an overlap concentration c* for the at least one framingassociative polymer relative to the host composition. In embodimentswhere the capping associative polymer is comprised in the non-polarcomposition, the capping associative polymer can be comprised in anamount up to 20% of a total associative polymer concentration of thenon-polar composition.In the associative non-polar composition herein described, when c≤2c*, μis μ_(h), and ρ is ρ_(h), and when c>2c*, μ is the viscosity of theassociative non-polar composition μ_(a), and ρ is the density of theassociative non-polar composition ρ_(a).

According to a fifth aspect a method is described, to control one ormore physical and/or chemical properties and in particular a rheologicalproperty of an associative non-polar composition in a flow characterizedby a Reynolds number Re, and a characteristic length d. The methodcomprises: providing a host composition having a viscosity μ_(h), adensity ρ_(h) and a dielectric constant equal to or less than about 5,and providing at least one framing associative polymer herein describedsubstantially soluble in the host composition and optionally at leastone capping associative polymer herein described.

In particular, in the method, the longest span of the at least oneframing associative polymer has a countour length ½L_(bf)≤L_(f)<L_(bf),wherein L_(bf) is a rupture length of the at least one framingassociative polymer in nanometers when the at least one framingassociative polymer is within the host non-polar composition at aconcentration c to provide the associative non-polar composition in aflow, L_(b) being given by implicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.The method further comprises combining the host composition and the atleast one framing associative polymer herein described at a selectedconcentration c between from about 0.01 c* to 10c*, depending on theweight-average molecular weight and/or Radius of gyration of the atleast one framing associative polymer and on the physical and/orchemical property to be controlled.In the method herein described, when c≤2c*, μ is μ_(h), and ρ is ρ_(h),and when c>2c*, μ is the viscosity of the associative non-polarcomposition μ_(a), and ρ is the density of the associative non-polarcomposition ρ_(a).In embodiments where the capping associative polymer is provided, themethod further comprises combining the at least one capping associativepolymer in the non-polar composition in an amount up to 20% of a totalassociative polymer concentration of the non-polar composition.In the method combining the at least one framing associative polymer andoptionally the at least one capping associative polymer is performed toobtain the associative non-polar composition. The method also comprisesapplying forces to the associative non-polar composition to obtain aflow characterized by the Reynolds number Re, and the characteristiclength d.

According to a sixth aspect a method is described, to control resistanceto flow and/or to control flow rate enhancement of an associativenon-polar composition alone or in combination with control of anotherphysical and/or chemical property of the associative non-polarcomposition in a flow characterized by a Reynolds number Re, and acharacteristic length d. The method comprises: providing a hostcomposition having a viscosity μ_(h), a density ρ_(h) and a dielectricconstant equal to or less than about 5, and providing at least oneframing associative polymer herein described substantially soluble inthe host composition and optionally at least one capping associativepolymer herein described. In the method the framing associative polymerand the capping associative polymer having a weight-average molecularweight equal to or higher to 200,000 g/mol.

In particular, in the method, the longest span of the at least oneframing associative polymer has a countour length ½L_(bf)≤L_(f)<L_(bf),wherein L_(bf) is a rupture length of the at least one framingassociative polymer in nanometers when the at least one framingassociative polymer is within the host non-polar composition at aconcentration c to provide the associative non-polar composition in aflow, L_(bf) being given by implicit function

$F_{bf} = {\frac{{{\pi\mu}_{h}}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho_{h}d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which in which F_(bf) is the rupture force of the framing associativepolymer in nanonewtons, Re is the Reynolds number of the flow, d is thecharacteristic length of the flow in meters, μ_(h) is the viscosity ofthe host non-polar composition in Pa·s, and ρ_(h) is the density of thehost non-polar composition in kg/m³.The method further comprises combining the host composition and the atleast one framing associative polymer herein described at a selectedconcentration c between from about 0.01 c* to 1c*, depending on theweight-average molecular weight and/or Radius of gyration of the atleast one framing associative polymer and on the extent of dragreduction desired alone or in combination with another physical and/orchemical property to be controlled. In embodiments where the cappingassociative polymer is provided, the method further comprises combiningthe at least one capping associative polymer in the non-polarcomposition in an amount up to 20% of a total associative polymerconcentration of the non-polar composition. In the method combining theat least one farming associative polymer and optionally the at least onecapping associative polymer is performed to obtain the associativenon-polar composition. The method also comprises applying forces to thenon-polar composition to obtain a flow characterized by the Reynoldsnumber Re, and the characteristic length d.

According to a seventh aspect a method is described to control sizes,and/or to control distribution of sizes of the droplets of a fluid (e.g.a fluid mist) in an associative non-polar composition in a flowcharacterized by a Reynolds number Re, and a characteristic length d,alone or in combination with another physical and/or chemical propertyof the non-polar composition in the flow. The method comprises providinga host composition having a viscosity μ_(h), a density ρ_(h) and adielectric constant equal to or less than about 5 and providing at leastone framing associative polymer herein described and optionally at leastone capping associative polymer herein described. In the method, theframing associative polymer and the capping associative polymer aresubstantially soluble in the host composition and have a weight-averagemolecular weight equal to or higher to 60,000 g/mol and in particularequal to or higher to 400,000 g/mol.

In particular, in the method, the longest span of the at least oneframing associative polymer has a countour length ½L_(bf)≤L_(f)<L_(bf),wherein L_(bf) is a rupture length of the at least one framingassociative polymer in nanometers when the at least one framingassociative polymer is within the host non-polar composition at aconcentration c to provide the associative non-polar composition in aflow, L_(b) being given by implicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.The method further comprises combining the host composition and the atleast one framing associative polymer herein described at a selectedconcentration c between from about 0.05c* to 3c*, depending on theweight-average molecular weight and/or Radius of gyration of the atleast one framing associative polymer and on the another physical and/orchemical property to be controlled.In the method herein described, when c≤2c*, μ is μ_(h), and ρ is ρ_(h),and when c>2c*, μ is the viscosity of the associative non-polarcomposition μ_(a), and ρ is the density of the associative non-polarcomposition ρ_(a).In embodiments where the capping associative polymer is provided, themethod further comprises combining the at least one capping associativepolymer in the non-polar composition in an amount up to 20% of a totalassociative polymer concentration of the non-polar composition. In themethod combining the at least one farming associative polymer andoptionally the at least one capping associative polymer is performed toobtain the non-polar composition.The method also comprises applying forces to the non-polar compositionto obtain a flow characterized by the Reynolds number Re, and thecharacteristic length d.

According to an eighth aspect, a method to provide an associativepolymer is described. The method comprises providing a linear, branchedor hyperbranched polymer backbone substantially soluble in a non-polarcomposition, in particular a host non-polar composition, the polymerbackbone having at least two ends and having a weight-average molecularweight equal to or higher than about 60,000 g/mol and in particularequal to or higher than 100,000 g/mol wherein a longest span of theassociative polymer has a contour length L, such that ½L_(b)≤L<L_(b),wherein L_(b) is a rupture length of the associative polymer innanometers when the associative polymer is within the host non-polarcomposition having a framing associative polymer concentration c toprovide an associative non-polar composition in a flow, L_(b) beinggiven by implicit function

$F_{b} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{b} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{b} \right)}} \times 10^{- 9}}$

in which F_(b) is the rupture force of the associative polymer innanonewtons, Re is the Reynolds number, d is the characteristic lengthof the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.In embodiments wherein c≤2c*, μ is the viscosity of the host non-polarcomposition μ_(h), ρ is the density of the host non-polar compositionρ_(h). In embodiments when c>2c*, μ is the viscosity of the associativenon-polar composition μ_(a), and ρ is the density of the associativenon-polar composition ρ_(a).The method further comprises attaching an associative functional groupat one or more ends of the at least two ends of the backbone. Inparticular in embodiments where the attaching is performed at two ormore ends of the at least two ends of the linear, branched orhyperbranched backbone the method provides a framing associativepolymer. In some embodiments the associative polymer has an overallweight average molecular weight, M_(w), equal to or lower than about2,000,000 g/mol, and/or a Mw equal to or higher than about 100,000g/mol. In some embodiments, the associative polymer is a framingassociative polymer. In some embodiments, the associative polymer is acapping associative polymer.

According to a ninth aspect a system is described for controlling aphysical or chemical property, and in particular a rheological property,of an associative non-polar composition in a flow characterized by aReynolds number Re, and a characteristic length d, alone or incombination with another physical and/or chemical property, and inparticular a rheological property, of the non-polar composition in theflow. The system comprises at least two between at least one hostcomposition herein described having a viscosity μ_(h), a density ρ_(h)and a dielectric constant equal to or less than 5, and at least oneframing associative polymer herein described substantially soluble inthe host. In the system, the longest span of the framing associativepolymer has a countour length ½L_(bf)≤L_(f)<L_(bf), wherein L_(bf) is arupture length of the framing associative polymer in nanometers when theframing associative polymer is within the host non-polar composition ata concentration c, to provide an associative non-polar composition in aflow, and L_(bf) is given by implicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4{\rho d}^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

In which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.In embodiments wherein c≤2c*, μ is the viscosity of the host non-polarcomposition μ_(h), and ρ is the density of the host non-polarcomposition ρ_(h). In embodiments when c>2c*, μ is the viscosity of theassociative non-polar composition μ_(a), and ρ is the density of theassociative non-polar composition ρ_(a).In some embodiments, the system can further comprise at least onecapping associative polymer herein described.

Additional examples, aspects and applications concerning the associativepolymers and related compositions, methods and systems of the presentdisclosure are set forth in the present description and provisionalapplication incorporated herein by reference in its entirety, which areprovided by way of illustration and are not intended to be limiting.

In particular, in some embodiments, the additional examples, aspects andapplications are related to polymeric fuel additives that can increasethe resistance to elongational deformation for a non-polar compositionand can reduce particulate emissions from engines.

Low concentrations of relatively high molecular weight polymers, such ashigh molecular weight polyisobutylene, are known as anti-mistingadditives. It is known that fuel-soluble high molecular weightpolyalphaolefins can improve fire safety and reduced risk of explosivecombustion of post-impact fuel mist. More recently, another benefit ofhigh molecular weight polyisobutylene (greater than about 4,000 kg/mol)in fuel was discovered, that is, improved combustion efficiency. [6]Widespread application of high molecular weight polymers in fuel hasbeen challenging in particular when maintenance of efficacy duringroutine fuel handling is desired. Passage through pumps, filters andpipelines breaks the polymer backbone. As the average length of thepolymer decreases, the effects associated with the presence of polymerscan be reduced. This phenomenon is known as shear degradation.

In some embodiments, associative polymers herein described comprisingpolymer chains that are individually short enough to resist sheardegradation and that have associative functional groups of appropriatestrength at appropriate positions on the polymer chain can reduce andeven minimize the shear degradation. In some embodiments individualpolymers reversibly assemble “mega-supramolecules” that confer thebenefits of high molecular weight linear polymers while greatly reducingor eliminating shear degradation. Like long polyisubutylene, themega-supramolecules resulting from the reversible assembly of theassociative polymers in a non-polar host composition can be sufficientlylarge that they are capable of carrying tensile stresses associated withan extensional or elongational force applied to the composition,resulting in an increased resistance to elongational deformation for thenon-polar composition.

One of the effects described herein enables the composition to formstable jet and/or filaments when subjected to elongational deformation.Another benefit provided by such associative polymers is that theyreduce soot formation when the fuel treated with the associativepolymers is burned in an engine. Although the mechanism for sootreduction by high molecular weight polymers in fuel is not known, it isexpected at least in some embodiments to occur through mist control.Specifically, it is expected that the enhanced elongation viscosityprovided by the polymer suppresses small satellite droplets.

The associative polymers, capping associative polymers and relatedmaterial compositions, methods and systems herein described can be usedin connection with applications wherein control of physical and/orchemical properties of non-polar compositions is desired with particularreference to drag reduction and/or flow rate enhancement. Exemplaryapplications comprise fuels and more particularly crude oils and refinedfuels, inks, paints, cutting fluids, drugs, lubricants, pesticides andherbicides as well as synthetic blood, adhesive processing aids,personal care products (e.g. massage oils or other non-aqueouscompositions) and additional applications which are identifiable by askilled person. Additional applications comprise industrial processes inwhich reduction of flow resistance, mist control, lubrication, and/orcontrol of viscoelastic properties of a non-polar composition and inparticular a liquid non polar composition is desired.

The details of one or more embodiments of the disclosure are set forthin the accompanying drawings and the description below. Other features,objects, and advantages will be apparent from the description anddrawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated into and constitute apart of this specification, illustrate one or more embodiments of thepresent disclosure and, together with the detailed description and theexamples, serve to explain the principles and implementations of thedisclosure.

FIGS. 1A-1B show a schematic illustration of supramolecular structuresof associative polymers according to embodiments herein described. Inparticular, FIG. 1A shows schematics of telechelic donor/acceptorinteraction. FIG. 1B shows schematics of telechelic self-associatinginteractions.

FIGS. 2A-2B show a schematic illustration of end to end association inassociative polymers herein described. FIG. 2A describes an exemplarydonor acceptor association FIG. 2B describes an exemplaryself-association.

FIG. 3 shows an exemplary associative polymer according to an embodimentherein described. In the illustration of FIGS. 3x and y can beindependently selected between any integer ≥1. The sum of x and y can bebetween 1,000 and 10,000.

FIG. 4 shows exemplary functional groups and related exemplaryassociative interactions according to embodiments herein described.

FIG. 5 shows exemplary architectures of associative polymers hereindescribed. In particular in the illustration of FIGS. 5,a, b, c, d, n,and e are independently integers ≥1.

FIG. 6 shows exemplary block architectures of associative polymersherein described and of an exemplary chain or backbone moiety. Inparticular in the illustration of FIGS. 6,a, b, c, d, n, x, and y areindependently integers ≥1.

FIG. 7 shows a schematic representation of a method to provide anassociative polymer of the disclosure according to embodiments hereindescribed.

FIG. 8 shows a schematic representation of a reaction suitable toprovide an associative polymer of the disclosure using chain transferagents according to embodiments herein described.

FIG. 9 shows exemplary chain transfer agents suitable to be used in thereaction illustrated in FIG. 8 according to embodiments hereindescribed, and in particular, chain transfer agents with internalolefins based on benzyl ether dendrons.

FIG. 10 shows a schematic representation of an exemplary method toproduce associative polymers herein described using chain transferagents according to embodiments herein described.

FIG. 11 shows a diagram illustrating GPC traces of 430K di-TE PB (di-TEPB also called octa tBu ester PB herein) and the resulting polymer ofits hydrolysis reaction (in TIF). In particular, FIG. 11, shows adiagram illustrating the GPC traces of a telechelic 1,4-PB with abackbone length of 430,000 g/mol (M_(w)) and end groups having 4tert-butyl ester groups on each (denoted TE groups hereinafter; thepolymer is denoted 430K di-TE PB hereinafter) and the resulting polymerof its hydrolysis reaction (in THF). The resulting end-groups with 4acid groups and the polymer are hereinafter denoted TA groups and 430Kdi-TA PB (di-TA PB also called octa acid PB herein), respectively.

FIG. 12 shows a diagram illustrating viscosity in function of shear rateof the 1 wt % Jet-A solutions of the 430K di-TE PB and 430K diTA PBherein also indicated as di-TE PB and (430K di-TA PB).

FIG. 13 shows a diagram illustrating GPC traces of the 430K octa chloroPB and the corresponding octa tertiary amine PB. In particular, FIG. 13,shows a diagram illustrating the GPC traces of telechelic 1,4-PB with abackbone length of 430,000 g/mol and end-groups with 4 chloro groups oneach and the corresponding tertiary amine-terminated polymer (the endgroups with 4 tertiary amines are denoted TB groups, and thecorresponding polymer is denoted 430K di-TB PB hereinafter).

FIG. 14 shows a diagram illustrating viscosity in function of shear rateof 1 wt % Jet-A solutions of 430K di-TE PB, di-TA PB, di-TA PB, and 1:1w/w di-TA PB/di-TB PB mixture herein also indicated as 430K di-TE PB,di-TA PB, di-TB PB, and 1:1 w/w -di-TA PB/di-TB PB mixture.

FIG. 15 illustrates properties of an exemplary hydrocarbon compositionaccording to the disclosure. In particular, FIG. 15, Panel A shows thatthe exemplary composition remains stable for months at −30° C. and FIG.15, Panel B shows that dewatering operations occur as quickly andcompletely in the composition (right) as in an untreated host (left).

FIG. 16 shows is a diagram illustrating the radius of gyration of anexemplary backbone polymer (polystyrene) as a function of itsweight-average molecular weight (M_(w) in g/mol) in a representativetheta solvent (cyclohexane) and in a representative good solvent(toluene). In particular, FIG. 16 shows an exemplary relationshipbetween the radius of gyration R_(g) of a backbone polymer as a functionof its weight average molecular weight (M_(w) in g/mol).

FIG. 17 shows a schematic representation of exemplary interactionsbetween conventional linear polymers of the disclosure, in situationwhen the polymer concentration is equal to the overlap concentration c*.The dotted lines represent the radius of the single polymers (functionalnot shown). In particular the schematic of FIG. 17, show an exemplaryway polymer molecules can pervade the entire solution when provided attheir overlap concentration c*.

FIGS. 18 and 19 show exemplary synthesis reactions for exemplary CTAssuitable to make associative polymers in accordance with embodimentsherein described.

FIGS. 20 and 21 show exemplary covalent links linking node to chain andnode to FG according to embodiments herein described.

FIG. 22 Shows a schematic illustration of the self-association behaviorof carboxyl-terminated telechelic 1,4-PBs according to some embodimentsherein described.

FIG. 23 shows a graph Specific viscosity of 1 wt % solutions of testpolymers in 1-chlorododecane (CDD) and tetralin (TL). FIG. 23, Panel Ashows the effect of end functionality N=1, 2, 4, 8 for polymers withM_(w)˜220,000 g/mol (Table 3.1). Data are not available forocta-carboxyl end groups (N=8) due to insolubility of the material inboth in CDD and TL. FIG. 23, Panel B shows results of N=4 at M_(w)=76,230 and 430,000 g/mol. Graphs are on different scales.

FIG. 24 shows the effect of number of chain-end functional groups (N) onthe concentration dependence of the specific viscosity of solutions oftelechelic associative polymers with M_(w)˜230,000 g/mol. FIG. 24, PanelA shows the effect in 1-chlorododecane (CDD).

FIG. 24, Panel B shows the effect in tetralin (TL). Graphs are ondifferent scales.

FIG. 25 shows the concentration dependence of specific viscosity ofsolutions of telechelic 1,4-PBs with non-associative and associativechain ends (N=4) as a function of M_(w): from left to right, 76,000g/mol, 230,000 g/mol, and 430,000 g/mol. The overlap concentration ofthe tertbutyl ester form of each polymer is indicated by the marks onthe concentration axis, circles and squares for tetralin (TL) andtriangles for 1-chlorododecane (CDD); for 76K di-TE in CDD c*=1.4 wt %(offscale). Solid lines indicate linear regression from 0.2 wt % to1.5c* for di-TE; dashed lines correspond to the solid line verticallyshifted to the linear portion of the di-TA data: red for TL and blue forCDD.

FIGS. 26A-26B show graphs depicting shear-thinning behavior of CDDsolutions and TL solutions. FIG. 26A shows CDD solutions of di-TA1,4-PBs at three concentrations (0.4, 0.7 and 1.0 wt %) as a function ofM_(w): FIG. 26A, Panel A 76,000 g/mol, FIG. 26A, Panel B M_(w)=230,000g/mol, and FIG. 26A, Panel C 430,000 g/mol. FIG. 26B shows TL solutionsof di-TA 1,4-PBs at three concentrations (0.4, 0.7 and 1.0 wt %) as afunction of M_(w): FIG. 26B, Panel A 76,000 g/mol, FIG. 26B, Panel BM_(w)=230,000 g/mol, and FIG. 26B, Panel C 430,000 g/mol.

FIG. 27 shows expanded ¹H NMR (500 MHz) spectra of CDCl₃ solutions oftelechelic polymers that have a 10,000 g/mol 1,4-PB backbone with endgroups. FIG. 27, Panel A shows the THY (thymine) spectrum. FIG. 27,Panel B shows DAAP (diacetamidopyridine). FIG. 27, Panel C shows thespectrum of a mixture of the two polymers with a mass ratio of 1:2,which represents a stoichiometric ratio of approximately 1:2. Theconcentration of polymer in solution is approximate 1 wt %.

FIG. 28 shows expanded ¹H NMR (500 MHz) spectra of CDCl₃ solutions oftelechelic polymers. FIG. 28, Panel A shows the spectrum of 1,4-PB ofM_(w)=50,000 g/mol with CA (cyanic acid) end groups FIG. 28, Panel Bshows the spectrum of 1,4-PB of M_(w)=24,000 g/mol with HR (Hamiltonreceptor) end groups. FIG. 28, Panel C shows a mixture of the twopolymers with a mass ratio of 1:1.4, which represents a stoichiometricratio of CA:HR of approximately 1:2. The concentration of polymer insolution is approximate 1 wt %.

FIG. 29 shows expanded ¹H NMR (500 MHz) spectra of CDCl₃ solutions oftelechelic polymers. FIG. 29, Panel A shows the spectrum of 1,4-PB ofM_(w)=22,000 g/mol with TB end groups. FIG. 29, Panel B shows thespectrum of a mixture of 1,4-PB of M_(w)=22,000 g/mol with TB end groupsand 1,4-PB of M_(w)=22,000 g/mol with TA end groups two polymers with amass ratio of 1:1. The concentration of polymer in solution isapproximate 1 wt %.

FIG. 30 shows expanded ¹H NMR (500 MHz) spectra of CDCl₃ solutions oftelechelic polymers. FIG. 30, Panel A shows the spectrum of 1,4-PB ofM_(w)=288,000 g/mol with THY end groups. FIG. 30, Panel B shows thespectrum of 1,4-PB of M_(w)=219,000 g/mol with DAAP end groups. FIG. 30,Panel C shows the spectrum of a mixture of the two polymers with a massratio of 1:2. The concentration of polymer in solution is approximate 1wt %.

FIG. 31 shows expanded ¹H NMR (500 MHz) spectra of CDCl₃ solutions oftelechelic polymers. FIG. 31, Panel A shows the spectrum of 1,4-PB ofM_(w)=200,000 g/mol with CA end groups. FIG. 31, Panel B shows thespectrum of 1,4-PB of M_(w)=240,000 g/mol with HR end groups. FIG. 31,Panel C shows the spectrum of a mixture of the two polymers with a massratio of 1:2. The concentration of polymer in solution is approximate 1wt %.

FIG. 32 shows expanded ¹H NMR (500 MHz) spectra of CDCl₃ solutions oftelechelic polymers. FIG. 32, Panel A shows the spectrum of 1,4-PB ofM_(w)=250,000 g/mol with TB end groups. FIG. 32, Panel B shows thespectrum of a mixture of 1,4-PB of M_(w)=250,000 g/mol with TB endgroups and 1,4-PB of M_(w)=230,000 g/mol with TA end groups two polymerswith a mass ratio of 1:1. The concentration of polymer in solution isapproximate 1 wt %.

FIG. 33 shows a plot of specific viscosity (25° C.) of 1 wt % CDDsolutions of 230K di-TE 1,4-PB, 230K di-TA 1,4-PB, 250K di-TB 1,4-PB,and the 1:1 (w/w) mixture of 230K di-TA 1,4-PB and 250K di-TB 1,4-PB atshear rates 1-3000 s⁻¹.

FIG. 34 shows a plot of specific viscosity (25° C.) of 1 wt % CDDsolutions of 230K di-DE 1,4-PB, 230K di-DA 1,4-PB, 250K di-DB 1,4-PB,and the 1:1 (w/w) mixture of 230K di-DA 1,4-PB and 250K di-DB 1,4-PB atshear rates 1-3000 s⁻¹.

FIG. 35 shows a plot of specific viscosity (25° C.) of 1 wt % Jet-Asolutions of 430K di-TE 1,4-PB, 430K di-TA 1,4-PB, 430K di-TB 1,4-PB,and the 1:1 (w/w) mixture of 430K di-TA 1,4-PB and 430K di-TB 1,4-PB atshear rates 1-3000 s⁻¹.

FIG. 36 shows GPC-LS (THF, 35° C.) traces of 230K di-TE 1,4-PB, 230Kdi-TA 1,4-PB and the resultant polymer of LAH reduction of 230K di-TA1,4-PB.

FIG. 37 shows a schematic illustration of a synthesis of di-TE 1,4-PBvia two-stage ROMP of COD as the benchmark reaction for the influence ofthe purity of VCH-free COD.

FIG. 38 shows a plot of the viscosities of a non-associative polymer inan appropriate host at varying concentrations using a rheometer whereinat c* a deviation from linearity is observed in the plot of viscosityversus polymer concentration. Linear regression is performed on the datafrom both dilute and concentrated regimes, and the crossover of the twolinear fits represents the overlap concentration, c*.

FIG. 39A shows an image of an experimental setup to test the associativepolymers herein described in the control of drag reduction incompositions (see, e.g. Example 13A).

FIG. 39B shows an image of an experimental setup to test the associativepolymers herein described in the control of long lasting drag reductionin compositions (see, e.g. Example 13B).

FIG. 39C shows that 1:1 (w/w) 670K Di-DA PB/630K Di-DB PB provideslong-lasting drag reduction.

FIG. 40 shows a plot of an exemplary relationship between c* and M_(w)that can be generalized to be used to select a desired M_(w) of abackbone in an associative polymer as herein described based on thedesired concentration of the associative polymer relative to c*.

FIG. 41 shows a schematic illustration of a two-stage synthesis oftert-butyl ester-terminated telechelic 1,4-PBs. Step (a): 50-100 equivof COD, 1/30 equiv of second-generation of Grubbs Catalyst, anhydrousdichloromethane (DCM), 40° C., 30-60 min. Step (b): 1000-2000 equiv ofCOD for target M_(w)<300,000 g/mol, anhydrous dichloromethane (DCM), 40°C., 16 h; 10000 equiv of COD for target M_(w)>400,000 g/mol, anhydrousdichloromethane (DCM), 40° C., <10 min.

FIG. 42 shows a schematic illustration of TFA hydrolysis of tert-butylester polymer end groups.

FIG. 43 shows graphs of specific viscosity (25° C.) of 1 wt %1-chlorododecane (CDD) and dodecane solutions of 288K di-THY 1,4-PB,219K di-DAAP 1,4-PB, and 1:2 (w/w) mixture of 288K di-THY 1,4-PB and219K di-DAAP 1,4-PB.

FIG. 44 shows a graph of Specific viscosity (25° C.) of 1 wt %1-chlorododecane (CDD) and Jet-A solutions of 240K di-HR 1,4-PB, 200Kdi-CA 1,4-PB, and 1:2 and 2:1 (w/w) mixtures of 240K di-HR 1,4-PB and200K di-CA 1,4-PB.

FIG. 45, Panels A-B show a schematic illustration of a synthesis ofdi-DB and di-TB 1,4-PBs via two-stage, post-polymerizationend-functionalization reaction.

FIGS. 46A-46B show a schematic representation of a synthesis ofbis-dendritic, tert-butyl ester-terminated chain transfer agents (CTA).FIG. 46A shows a synthesis of a CTA with only one tert-butyl ester oneach side (compound 3). FIG. 46B shows a synthesis of a CTA with onlyone tert-butyl ester on each side (compound 10), with the conditionsbeing: (a) 2.2 eq. of 2 or 2′, K₂CO₃, N,N-dimethylformamide (DMF), 80°C., 5 h; (b) 4 eq. of LiAlH₄, THF, R.T., overnight; (c) 6 eq. of 2 or2′, 6 eq. of PPh₃, 6 eq. of DIAD, THF, 0° C. then 40° C., overnight; (d)8 eq. of LiAlH₄, THF, R.T., overnight; (c) 12 eq. of 3, 12 eq. of PPh₃,12 eq. of DIAD, THF, 0° C. then 40° C., overnight.

FIGS. 47A-47C show assembly of long telechelic polymers (LTPs) intomega-supramolecules (right; linear and cyclic (not shown)) compared tothat of randomly functionalized associative polymers (left) and priorend-associative telechelics (middle) in terms of degree ofpolymerization (DP) and conformations at rest and in elongational flow;FIG. 47B ring-chain equilibrium distribution of cyclic (filled) andlinear (open) supramolecules; FIG. 47C synthesis of telechelics(non-associative with FG end-groups, structures in FIG. 61 and FIG. 45A)and post-polymerization conversion to associative telechelics (FGa,bottom). (1): Grubbs II, dichloromethane (DCM), 40° C., 1 h; (2): GrubbsII, DCM, 40° C., until stir bar stops (>5 min), equivalents of COD fordesired molecular weight. DA: di-acid. DB: di-base. TA: tetra-acid.

FIGS. 48A-48D show evidence of supramolecules in solutions of equimolarmixture of α,ω-di(isophthalic acid) and α,ω-di(di(tertiary amine))polycyclooctadienes (DA/DB); FIG. 48A effect of telechelics size(k≡kg/mol) on specific viscosity of supramolecular solutions andcontrols in cyclohexane (CH) at 2 mg/ml (0.25% wt, 25° C.); FIG. 48Beffect of solvent on specific viscosity for 2 mg/ml (0.25% wt) solutions(25° C.) of telechelics having M_(w)=670k due to both polarity(dielectric constant, FIG. 80) and solvent quality for the backbone(FIG. 63, Panel A); FIG. 48C, static light scattering (35° C.) showsthat association of ˜670k DA with DB chains in CH at 0.22 mg/ml (0.028%wt) produces supramolecules (filled) with an apparent M_(w) greater than2,000 kg/mol, which separate into individual building blocks (x) when anexcess of a small-molecule tertiary amine is added (open symbols, 10ul/ml of triethylamine, TEA; see FIG. 63, Panels A-B for its effect onviscosity). Curves show predictions of the model for complementarytelechelics 1,000 kg/mol in solution at 1400 ppm (solid, supramolecules;dashed, non-associated telechelics), details in FIG. 63, Panel C; FIG.63, Panel D concentration-normalized SANS intensities (25° C.) for 50ktelechelics in d12-cyclohexane at concentrations well below the overlapconcentration of NA (2 mg/ml for NA and DB; 0.05 mg/ml for DA andDA/DB).

FIGS. 49A-49C show the decrease of specific viscosity for 4.2M PIB 1.6mg/ml (0.2% wt) in Jet-A at 25° C. after approximately 60 passes througha Bosch fuel pump as shown in FIG. 66, Panel A (sheared) relative toas-prepared (unsheared) indicates shear degradation; FIG. 49B Specificviscosities of 2.4 mg/ml (0.3% wt) of a 1:1 molar ratio ofα,ω-di(isophthalic acid) and α,ω-di(di(tertiary amine))polycyclooctadienes (˜670 kg/mol DA/DB) in Jet-A at 25° C., sheared vs.unsheared; FIG. 49C Emission data using an unmodified long-haul dieselengine. Control: untreated diesel. Treated: diesel treated with 0.1% wt670k DA/DB (details in Example 63).

FIGS. 50A-50B show impact test in the presence of ignition sources (60ms after impact, maximal flame propagation) for Jet-A solutions treatedwith 4.2M PIB or α,ω-di(di-isophthalic acid) polycyclooctadienes (TA):FIG. 50A Jet-A with 4.2M PIB (0.35% wt) and Jet-A with 430k TA (0.3%wt), “unsheared” and “sheared”; FIG. 50B effect of TA molecular weight(76 kg/mol to 430 kg/mol) in Jet-A at 0.5% wt (unsheared).

FIG. 51 shows model predictions for two different values of the strengthof interaction εkT=14 kT (left), εkT=16 kT (middle) and εkT=18 kT(right) (open diamond: linear supramolecules; solid diamond: cyclicsupramolecules).

FIG. 52 shows molecular design for self-assembly of telechelic polymericbuilding blocks into larger linear and cyclic supramolecules via endassociation

FIG. 53 shows grouping of polymer components, where A and B genericallyrefer to A1 or A2 and B1 or B2 end-groups. Each group is composed of allthe different possible aggregates obtained by the assembly of the A1-A2and B1-B2 building blocks.

FIG. 54 shows mapping of polymer loops into necklaces of 4 colors. The 4colors correspond to: A1A2B1B2, A1A2B2B1, A2A1B1B2, A2A1B2B1.

FIG. 55 shows that it is not possible to create a loop that “reads” thesame clockwise and counterclockwise, so every loop maps into exactly twodistinct necklaces. (Color assignments are given in FIG. 54).

FIG. 56, Panels A-C show contact probabilities and equilibria.

FIG. 57 shows selection of the end-groups; FIG. 57, Panel A chemicalstructures and molar masses of the end-associative polymers (exceptingisophthalic acid/tertiary amine functionalized ones that are shown inFIG. 47C); FIG. 57, Panel B specific viscosities of telechelic polymersat 8.7 mg/ml total polymer in 1-chlorododecane; FIG. 57, Panel Cillustration of secondary electrostatic interactions (SEIs) in THY/DAAPand HR/CA pair.

FIGS. 58 and 59 show incorporation of CTA into polymer during the firststage of two-stage ROMP of COD, and chain extension to long telechelicsin the second stage: FIG. 58, ¹H NMR of characteristic peaks fordi(di-tert-butyl-isophthalate) CTA (structure of end-group shown in FIG.57), unreacted CTA (proton 1) and CTA incorporated into macromer (proton2), at three time points; the integrations of the peaks were used tocalculate the percentage of unreacted CTA, shown in part FIG. 59, PanelA. FIG. 59, Panel A, Kinetic curves show that the peaks characteristicof the unincorporated CTA are already difficult to quantify in thesample taken after 40 min, and it is not evident for the sample taken at1 hour (given the magnitude of the noise in the spectra, the amount ofunincorporated CTA is less than 3%). Dashed curve is calculated basedthe data point at 10 min assuming exponential decay of unreacted CTA.FIG. 59, Panel B, In an example with di-chloro PCOD, the M_(n)calculated by NMR is in good agreement with that measured by GPC,considering the inherent uncertainty in NMR integration and the inherentuncertainty in GPC measurement (5-10%). FIG. 59, Panel C, GPC tracesshow no indication of macro CTA (42 kg/mol) in the chain-extendedtelechelics (structure shown in FIG. 59, Panel C, 497 kg/mol) producedin the second step.

FIG. 60 shows ¹H NMR spectra of increasingly purified COD in the rangefrom 3.4 to 5.9 ppm: FIG. 60, Panel A COD after BH₃.THF treatment andvacuum distillation (containing ˜330 ppm of butanol based onintegration); FIG. 60, Panel B COD further purified with magnesiumsilicate/CaH₂ treatments (to show removal of butanol and the resultingpurity of COD used as monomer).

FIGS. 61A-61B show structures of non-associative (NA) end-groups and theconversion from NA to associative end-groups; FIG. 61B, isophthalic acidend groups obtained by deprotection of the tBu groups in thetBu-ester-ended non-associative precursor.

FIGS. 62A-62B show ¹H NMR spectra of tBu-ester ended (DE) andisophthalic acid ended (DA) polycyclooctadiene (M_(w)=630 kg/mol)showing high degree of conversion of the end-groups: FIG. 62A, the peaksfor protons on the phenyl ring (protons 1 and 2) shift due to theremoval of tBu; FIG. 62B, the peak for tBu group disappears in thespectrum for DA.

FIG. 62C shows ¹H NMR spectra of azide ended (DN₃) and tertiary amineended (DB) polycyclooctadiene (M_(w)=540 kg/mol) showing high degree ofconversion of the end-groups.

FIG. 63 shows formation of supramolecules in equimolar solutions ofα,ω-di(isophthalic acid) polycyclooctadiene, α,ω-di(di(tertiary amine))polycyclooctadiene (DA/DB), with non-associated controls: FIG. 63, PanelA, effect of chain length on specific viscosity of telechelics intetralin and Jet-A (2 mg/ml) at 25° C.; FIG. 63, Panel B, effect of TEA(2.5 μl/ml) on the viscosities of associative telechelic polymers DA/DB;FIG. 63, Panel C, left: static light scattering shows that associationbetween DA and DB chains (circle: 670k series; triangle: 300k series) inCH at 0.22 mg/ml (0.028%) produces supramolecules (filled), whichseparate into individual building blocks (x) when an excess of asmall-molecule tertiary amine is added (open symbols, 10 μl/ml oftriethylamine, TEA). Curves show predictions of the model (see Examples37-49); right: Zimm plot of the same static light scattering data shownin Left part. Lines indicate the fitting to the Zimm equation and dashedlines indicate the extrapolation that was used to evaluate the interceptat zero concentration, zero angle; the slope of the line and the valueof the intercept are used to evaluate the apparent M_(w) and apparentR_(g); FIG. 63, Panel D, resulting values of apparent M_(w) and R_(g)for the five polymer solutions in FIG. 63, Panel C.

FIG. 64 shows modeling of interplay of telechelic length andconcentration in a stoichiometric mixture of complementaryend-associative telechelics in the regime of long telechelics: FIG. 64,Panel A, effect of telechelic length on the distribution of the numberof telechelics in a supramolecule, given as the concentration in ppmwt/wt of each species, cyclic (circles) or linear (x or +), at a fixedtotal concentration of 1400 ppm; FIG. 64, Panel B the same distributionsas in FIG. 64, Panel A, presented in terms of the molar mass of thesupramolecules; the weight-average molar mass of the supramolecules isgiven to the left of the legend; FIG. 64, Panel C effect ofconcentration on the distribution of supramolecules for telechelics of1M g/mol (see Examples 37-49).

FIG. 65 shows ¹H NMR spectra of isophthalic acid ended (DA) anddi(tertiary amine) ended (DB) polycyclooctadienes (M_(w)=45 kg/mol) and1:1 molar mixture of DA/DB in deuterated chloroform (CDCl₃) indicatingthat carboxylic acid-amine hydrogen bonds dominate over carboxylicacid-carboxylic acid hydrogen bonds: FIG. 65, Panel A, ¹H NMR peaks dueto hydrogens on carbons adjacent to nitrogens of tertiary amine groupsof DB (methylene protons 1; methyl protons 2) shift downfield when theyform charge-assisted hydrogen bonds with carboxylic acid groups of DA;FIG. 65, Panel B, ¹H NMR peaks due to hydrogens on the phenyl ring of DAshift upfield upon formation of charge-assisted hydrogen bonds betweencarboxylic acids and tertiary amines.

FIG. 66 shows: FIG. 66, Panel A home-built apparatus for “sheardegradation” test; FIG. 66, Panel B an initially 4,200 kg/mol PIB at aconcentration of 0.35% in Jet-A shows the decrease in specific viscosityindicative of shear degradation with increasing number of passes throughthe pump; FIG. 66, Panel C, GPC validation of “shear degradation” testusing PIB and confirmation that associative polymers resist degradation(see Example 61).

FIG. 67A shows results of diesel engine tests using The Federal TestProtocol (FTP) with a specified transient of RPM and torque designed toinclude segments characteristic of two major cities (NY and LA); FIG.67B shows work and fuel efficiency data using an unmodified long-hauldiesel engine. Control: untreated diesel. Treated: diesel with 0.14% w/v670k DA/DB (see Examples 63 and 64).

FIG. 68 shows average mass flow rate normalized to that of “as prepared”4.2M PIB solution for a 0.02% solution of 4.2M PIB in Jet-A and a 0.1%solution of 670k DA/DB in Jet-A (similar to that used in the dieselengine tests of FIG. 49C).

FIG. 69 shows FIG. 69, Panel A apparatus for impact/flame propagationexperiments; FIG. 69, Panel B frame at 60.4 ms for untreated Jet-A. Therectangular box is the area within which pixels were analyzed forbrightness; FIG. 69, Panel C average brightness of the pixels in therectangle of FIG. 69, Panel B as a function of time during the first 300ms after impact for five compositions (untreated Jet-A, 0.35% wt 4.2MPIB unsheared, 0.35% wt 4.2M PIB sheared, 0.3% wt 430k TA unsheared and0.3% wt 430k TA sheared).

FIG. 70 shows characterization of α,ω-di(di(isophthalic acid)) (TA)polycyclooctadiene used in Impact test: FIG. 70, Panel A, Effect ofchain length (k refers to kg/mol) on specific viscosity of TA intetralin at 10 mg/ml. FIG. 70, Panel B Specific viscosity of 2.4 mg/ml430k TA in Jet-A at 25° C., sheared vs unsheared.

FIG. 71 shows a schematic representation of the concentration-dependentself-association of telechelic associative polymers (see FIG. 1B). Left:Telechelic associative chain at low concentration. Middle: Flower-likemicelle above a critical concentration value. Right: Transient networkat higher concentration.

FIG. 72 shows specific viscosity of 1 wt % Jet-A solutions of LTPs at25° C.: FIG. 72, Panel A, 430 kg/mol NA-, TA-, TB-PCODs, and 1:1 (w/w)mixture of TA- and TB-PCODs; FIG. 72, Panel B, 200 kg/mol NA-, DA-,DB-PCODs, and 1:1 (w/w) mixture of DA- and DB-PCODs; FIG. 72, Panel C,600 kg/mol NA-, DA-, DB-PCODs, and 1:1 (w/w) mixture of DA- andDB-PCODs. Note that all data reported are averages over shear rates 10to 100 s.

FIG. 73 shows representative examples of solutions of associative LTPsin Jet-A after storage at −30° C. over 13 months: 0.3 wt % Jet-Asolution 1:1 (w/w) mixture of 430 kg/mol TA- and TB-PCODs. (See FIG. 15,Panel A (left panel) for 0.5 wt % Jet-A solution of 264 kg/mol TA-PCOD).

FIG. 74 shows shear viscosity of samples from shear stability test andtheir unsheared controls. Right: 0.35 wt % Jet-A solution of 4,200kg/mol PIB; middle: 0.3 wt % Jet-A solution of 430 kg/mol TA-PCOD; left:0.3 wt % Jet-A solution of 1:1 mixture of 600 kg/mol DA- and DB-PCODs.

FIG. 75, Panels A-B shows results of Jet-A in impact/flame propagationtest: FIG. 75, Panel A t=30 ms after impact FIG. 75, Panel B t=60 msafter impact.

FIG. 76 shows results of 0.35 wt % Jet-A solution of 4,200 kg/mol PIB inimpact/flame propagation test. Left: results of unsheared solution;right: results of sheared solution.

FIG. 77 shows results of 0.3 wt % Jet-A solution of 430 kg/mol TA-PCODin flame propagation test. Left: results of unsheared solution. Right:results of sheared solution.

FIG. 78 shows molecular design considerations for backbone selection forsolubility in fuels and resistance to chain scission. In contrast topolymers examined in prior literature ([7],[8]) on mist control and dragreduction, the present polymers use a backbone that has no tertiary orquaternary carbons nor any heteroatoms in the repeat unit. Theimportance of these features is illustrated by comparison with the twopolymers that have received the most attention in prior literature:4,200 kg/mol polyisobutylene (PIB) and a copolymer of acrylic andstyrenic monomers known as FM-9 (M_(w)˜3,000 kg/mol). Acrylate unitsintroduce heteroatoms that interfere with fuel solubility (a problemthat is exacerbated by the random incorporation of carboxylic acid sidegroups). Polyisobutylene has quaternary carbons in the backbone, makingit particularly susceptible to chain scission ([9]). The tertiarybackbone carbons in FM-9 also make the backbone more susceptible tochain scission than one that has only secondary carbons. The solubilityand strength of the present polymers are enhanced by includingcarbon-carbon double bonds in the backbone.

FIG. 79 shows physical properties of single component solvents:Dielectric constant (ε) and refractive index (n). Dielectric constantserves as a measure of the polarity of solvents: it increases from forcyclohexane (CH) and tetralin. Increasing solvent polarity reduces thedegree of end-association for the telechelics. The difference betweenthe refractive index of solvents and that of PCOD (n˜1.52) determinesthe contrast in multi-angle laser light scattering (MALLS). Tetralin isexcluded from the MALLS experiment because of its low contrast with PCOD(1.54 is too close to 1.52). Cyclohexane gives desirable contrast inMALLS.

FIG. 80 shows preliminary ASTM data of untreated (“Base fuel”) andtreated JP-8 (with 1:1 molar mixture of 500 kg/mol α,ω-di(isophthalicacid) polycyclooctadiene and 600 kg/mol α,ω-di(di(tertiary amine))polycyclooctadiene (DA/DB)). ^((a))The concentration of polymer(mass/mass) added to “Base fuel” JP-8, a military aviation fuel(specified by MIL-DTL-83133), corresponding to Jet-A with threeadditional additives: the Corrosion Inhibitor/Lubricity Enhancer, theFuel System Icing Inhibitor, and the Static Dissipater Additive.^((b))Flash Point (ASTM D93) is the lowest temperature at which fuelwill produce enough flammable vapors to ignite when an ignition sourceis applied. Flash point is the most commonly used property for theevaluation of the flammability hazard of fuels. As expected, themist-control polymers do not affect the flash point because the polymeradditive affects mechanical mist formation—not the liquid-vaporequilibrium characteristics of the fuel. There is no statisticallysignificant difference in flash point among the three samples.^((c))Total Acid Number (ASTM D3242) organic acids are naturally foundin hydrocarbon fuels and others are created during refining. Thepresence of acids in fuel is unwanted because of the potential to causecorrosions or interfere with fuels water separation. There is nostatistically significant difference in total acid number among thethree samples. ^((d)) Density at −15° C. (ASTM D4052) is used to verifyfuel type, calculate aircraft fuel load and range, gaging and meteringand flow calculations. (e) Kinematic Viscosity at −20° C. (ASTM D445) atlow temperatures is specified to be 8.0 mm²/s or less to ensure adequatefuel flow and atomization under low temperature operations, particularlyfor engine relight at altitude. The composition at 1000 ppm obeys thiscriteria.

FIGS. 81A-81H show associative polymer based on 2-arm linear (e.g. FIG.81A) and 3-arm star structure units (e.g. FIG. 81B) in which each chainis connected to a least one node “N”. Within the class of molecules asdescribed herein; in strong flow the molecule is expected to tend tobreak near the middle or a node “N”, so one of the two resultant piecesmay retain at least end functional groups and efficacy is expected toremain substantially unchanged (e.g. FIG. 81F). In an H shaped polymeras shown in the molecule (e.g. FIG. 81H) breaks near the middle;resulting in two polymers (each half of the H polymer which arethemselves active for the desired rheological effect, so efficacy is notlost.

FIG. 82 shows a table indicating values of viscosities for exemplaryhost composition liquids at a pressure of 1 atm and at a temperature of300 K (27° C.).

FIG. 83 shows a table indicating experimental density and viscosity ofexemplary composition liquids at a pressure of 1 atm as a function oftemperature.

FIG. 84 shows a table indicating average bond enthalpies (kJ/mol) ofcovalent bonds including single bond and multiple bonds.

FIG. 85 shows a chart illustrating a graphic solution of equation

$F_{k} = \frac{{\pi\mu}^{2}{Re}^{3/2}L^{2}}{\left. {4{\rho d}^{2}{\ln\left( {{L/1}\mspace{14mu}{nm}} \right)}} \right)}$

[nN] versa L [nm] in which is F_(k) is Kolmogorov force of a non-polarcomposition exerting hydrodynamic forces on an associative polymer inthe composition.

FIG. 86 shows a graph indicating the combination of variables computedfrom the observed length of chains after hydrodynamic scission at aparticular Reynolds number, the viscosity and density of the exemplaryhost composition and the characteristic length d of the flow as afunction of the Reynolds number. The equation for the hydrodynamictension is shown in the insert. PS is polystyrene (in decalin ortoluene). PEO is polyethylene oxide (in water). PAM is polyacrylamide(in water). CS indicates a cross-slot flow. CE indicates acontraction/expansion flow. RT indicates a rotational turbulent flow.L_(b) is the contour length corresponding to the weight-averagemolecular weight of the chains after the flow experiment. a is 1 nm. μis the dynamic viscosity of the host composition. ρ is the density ofthe solvent. d is the characteristic length of the flow (for CS,d=channel width; for CE, d=diameter of the orifice; for RT, d=gapbetween the moving surfaces). Re is the Reynolds number of the flow.F_(K) is the hydrodynamic force at the Kolmogorov length scale for aslender rod of length L.

FIG. 87 Panel (A) shows the structure of a three-arm polymer having anisocyanurate node and three FGa-chain-units which contain m, p and qrepeat units respectively with the longest span emphasized in bold; FIG.87 Panel (B) shows the structure of a linear polymer with anisocyanurate node and two FGa-chain-units, which contain p and q repeatunits respectively with the longest span emphasized in bold.

FIG. 88 Panel (A) shows the structure of a three-arm polymer having atrioxymethyl ethane node and three chain units, which contains m, p andq repeating units respectively with the longest span emphasized in bold,where * is an associative functional group FGa as disclosed herein; FIG.88 Panel (B) shows the structure of a linear polymer having atrioxymethyl ethane node and two chain units, which contains p and qrepeating units respectively with the longest span emphasized in bold,where * is an associative functional group FGa as disclosed herein.

FIG. 89 Panel (A) shows the structure of a polystyrene polymer (PS)having m repeat units and a corresponding 2m number of C—C backboneatoms, with the longest span emphasized in bold, where * is anassociative functional group FGa as disclosed herein; FIG. 89 Panel (B)the structure of a PS-co-PSBr statistical copolymer having p styreneunits and q bromostyrene units and a corresponding 2(p+q) number of C—Cbackbone atoms, with the longest span emphasized in bold, where * is anassociative functional group FGa as disclosed herein.

FIG. 90 Panel (A) shows the structure of a FGa-chain-FGa statisticalco-polymer having p norbornene imide units and q norbornene diesterunits and a corresponding 5(p+q) total number of backbone atoms; FIG. 89Panel (B) shows the structure of a FGa-chain-FGa statistical co-polymerhaving p norbornene imide units and q norbornene diester units and acorresponding 5(p+q) total number of backbone atoms.

FIG. 91 Panel (A) shows a schematic of the construction of thecross-slot flow cell;

FIG. 91 Panel (B) shows the top view of the central block; FIG. 91 Panel(C) shows the flow arrangements for high strain rate experiments by Xueet al. [10].

FIG. 92 Panel (A) shows the tandem GPC-MALLS characterization resultsobtained for polystyrene (starting M_(w)=8470 kg/mol and M_(n)=3940kg/mol) at 100 ppm (w/v) in decalin at Reynolds number of 29,400; FIG.92 Panel (B) shows the tandem GPC-MALLS characterization resultsobtained for polystyrene (starting M_(w)=8470 kg/mol and M_(n)=3940kg/mol) at 100 ppm (w/v) in decalin at Reynolds number of 4,290; FIG. 92Panel (C) shows the tandem GPC-MALLS characterization results obtainedfor polystyrene (starting M_(w)=8470 kg/mol and M_(n)=3940 kg/mol) at100 ppm (w/v) in decalin at Reynolds number of 2,590.

DETAILED DESCRIPTION

Associative polymers, and related materials, compositions, methods, andsystems are described, which based in several embodiments, allow controlof physical and/or chemical properties, of a non-polar composition in aflow.

“Chemical and/or physical properties” in the sense of the presentdisclosure comprise properties that are measurable whose value describesa state of a physical system and any quality that can be establishedonly by changing a substance's chemical identity.

The term “non-polar compositions” in the sense of the present disclosureindicates compositions having a dielectric constant equal to or lowerthan 5 which can comprise compositions of varying chemical nature. Inparticular, a non-polar composition can comprise hydrocarboncompositions, fluorocarbon compositions or silicone compositions. Ahydrocarbon composition is a composition in which the majority componentis formed by one or more hydrocarbons. A fluorocarbon composition is acomposition in which the majority component is formed by one or morefluorocarbons. A silicone composition is a composition in which themajority component is formed by one or more silicones. Accordingly, acomposition in the sense of the present disclosure can comprise onecomponent (e.g. a non-polar solvent) and traces of additional components(such as additives and/or preservative of the solvent).

Non-polar composition herein described comprise host non-polarcomposition (or host composition) typically provided by a liquidsolvent, and associative non-polar compositions which typically comprisea host composition and one or more associative polymers hereindescribed.

Non polar composition herein described can be characterized bycomposition density ρ and viscosity μ.

In particular the density ρ is a volumetric mass density of a non-polarcomposition which is defined as a mass of the non-polar composition perunit volume. The mass of the non-polar composition can normally bemeasured with a scale or balance; the volume of the non-polarcomposition can be measured directly graduated vessel. Alternativelymethods and devices to determine the density of a liquid, comprise ahydrometer, or a Coriolis flow meter and additional devices as will beunderstood by a skilled person. The density of a host composition isherein also indicated as ρ_(h). The density of an associative non-polarcomposition is herein also indicated as ρ_(a).

The viscosity μ is a measure of resistance of a liquid to gradualdeformation by shear stress, such as the shear stress in a liquid undervarious flow conditions. Viscosity μ can be measured with various typesof viscometers and rheometer as will be understood by a skilled person.In host composition the viscosity ρ_(h) of the host composition is theshear viscosity η_(s) or η_(solvent) of the host composition which is ameasure of resistance of the host composition to shearing flows, whereadjacent layers move parallel to each other with different speeds. (seeExamples 16-17).

In associative non polar composition the viscosity μ_(a) of theassociative non-polar composition is μ_(a)=η_(solution) whereinη_(solution) is the viscosity of the associative non-polar composition,that can be experimentally measured. η_(solution) can be used to deriveadditional parameters defining the contribution of associative polymersto the viscosity of the associative non-polar composition by

η_(sp),≡(η_(solution)−η_(solvent))/η_(solvent).

wherein specific viscosity η_(sp) as used herein is defined as a ratiothe change in viscosity of a liquid host composition (for example asolvent) due to the presence of a solute such as a polymer to theviscosity of the liquid host in the absence of the solute.

Associative non-polar composition can also be characterized by anextentional viscosity or elongational viscosity η_(ext), which is ameasure of the resistance to the “pull” (or more specifically,extensional or elongational deformation) placed on a liquid. Theextensional viscosity of a liquid tells how difficult it is to stretch athread of such a liquid. A skilled person in the art can understand thatin uniaxial extension, the extensional viscosity is defined as the ratioof the difference between axial and radial normal stresses, to the rateof axial extensional deformation:[11]

$\eta_{ext} = \frac{\tau_{zz} - \tau_{rr}}{\overset{.}{ɛ}}$

Where η_(ext) is extensional viscosity, τ_(zz) is the axial normalstress, τ_(rr) is the radial normal stress, and {dot over (ε)} is therate of axial extensional deformation.

Experimentally, extensional viscosity η_(ext) can be measured bytechniques identifiable to a skilled person, such as opposed nozzle,[12,13] entry flow,[14] and capillary break-up elongational rheometry(CaBER).[15] Among these methods, CaBER is popular among those skilledin the art because (1) the experimental setup is easy to handle, (2)small amount of sample is needed for the experiment, (3) it isapplicable for a wide range of shear viscosity (0.05-10 Pa·s), and (4)it is capable of generating large extensional strains.[16] In CaBER,what is measured is the time-evolution of the diameter of the filamentat the midpoint resulting from stretching, and the apparent extensionalviscosity of the fluid sample is determined using the following formula:

$\eta_{ext}^{app} = \frac{\sigma}{{{dD}_{mid}(t)}/{dt}}$

where η_(ext) ^(app) is the apparent extensional viscosity of the fluidsample, a is the surface tension of the fluid sample, and dD_(mid)(t)/dtis the rate of change in diameter of the filament at midpoint.Techniques to measure a are identifiable to a skilled person, anddD_(mid)(t)/dt is monitored and recorded by a laser micrometer connectedto a computer on which software processes the data and calculate η_(ext)^(app) as a function of extensional strain rate, {dot over (ε)}, basedon other physical parameters of the test fluid identifiable to a skilledperson.

The extensional viscosity, or the resistance to elongational deformationfor a polymer solution, is dictated by the size of the polymer (in termsof molecular weight). Gupta and co-workers found that for a dilutesolution (i.e., c<c*), before the elongation of the fluid reaches itssteady-state asymptote the measured apparent extensional viscosityη_(ext) ^(app) (called “transient extensional viscosity”) scalesc¹·M_(w) ¹.[17] As for the asymptotic behavior of a fluid in extensionalflow, the steady-state extensional viscosity shows a strong dependenceof polymer M_(w), as depicted in the following scaling relationship:[15]

(η_(ext) ^(∞)−3·η_(s))˜M _(w) ^(v+1)

Where η_(ext) ^(∞) is the steady-state extensional viscosity of thefluid, η_(s) is the shear viscosity of the solvent (or host), and v isthe excluded volume parameter.

In embodiments herein described, the non-polar composition and inparticular the associative non-polar composition is in a flow. A flow asused herein refers a movement of a continuum of liquid with unbrokencontinuity. The flow of liquid is governed by basic physical laws ofconservation of mass, momentum and energy. The properties of a flowproperties include flow velocity, pressure, density, viscosity, storageand loss moduli, viscoelasticity, and temperature, as functions of spaceand time.

Liquid flow can be in the form of a laminar flow and a turbulent flow. Aturbulent flow is characterized by recirculation, eddies, and apparentrandomness. A turbulent flow tends to produce chaotic eddies, vorticesand other flow instabilities. In contrast, a laminar flow is a movementof a liquid in parallel layers, with no disruption between the layers. Alaminar flow is characterized by smooth, constant fluid motion. Influids comprising a polymer, the hydrodynamic force is proportional tothe viscosity times the local elongational strain rate times the squareof the contour length of the longest span of the polymer. In a steadylaminar flow the elongation rate typically varies with position and issteady with time. In a turbulent flow, the strain rate is not uniform intime or space.

A flow can be defined by various characterizing features identifiable bya skilled person. In particular flows in non-polar composition hereindescribed can be characterized by Reynolds number Re and acharacteristic length d.

A Reynolds number as used herein indicates a dimensionless quantity thatcharacterizes the ratio of fluid inertial effects to viscous effects ina specific flow condition of a specific liquid. Reynolds number can beused to identify a type of flow. For example, a flow that has a Re<2000a flow is considered laminar, while a flow with Re>5000 a flow isconsidered turbulent. Reynolds number can be calculated based on thedensity and viscosity of the liquid, as long as the velocity of the flowand the characteristic length of the flow d are known as will beunderstood by a skilled person.

A characteristic length d is a lateral dimension of a bounded volume ofa liquid through which the liquid flows. The characteristic length isgiven by four times the ratio of the cross sectional area orthogonal tothe prevailing flow direction to the perimeter of the bound volume ofliquid. For a flow through an orifice, the length scale is the diameteror hydraulic diameter of the orifice through which the liquid flows. Fora flow through a circular tube or a conduit, the characteristic lengthis the diameter of the circular tube or a conduit.

A transition from laminar to turbulent flow of Newtonian liquids can bepredicted by the value of the Reynolds number at which the transitionoccurs. For example, in straight conduits with uniform cross section(such as pipes and ducts), the transition value of the Reynolds numberis approximately Re=2100. When irregularities are present in the path ofthe fluid the Reynolds number at the transition to turbulence, Re, isreduced relative to the value for straight conduits as will beunderstood by a skilled person. In flows through a conduit with arelatively steep increase in the cross section, the transition toturbulence can occur for example at Re=370. In a flow with a stagnationpoint, such as T-junction or a cross junction (two oppositely directedstreams coming together and two oppositely directed streams leaving thejunction), laminar flow can become unstable and transitions toturbulence for example at Re=25. A skilled person can calculate theReynolds number of the flows that are present in a specific applicationin which they intend to use the associative polymers herein described.

In embodiments herein described, associative polymers are provided whichcan be added to a non-polar composition to control at least one physicaland/or chemical property of the composition as illustrated in thepresent disclosure. In particular, chemical and/or physical propertiesthat can be controlled by associative polymers herein described includerheological properties. The term “rheological properties” of acomposition refers to properties related to the deformation and flow ofthe composition, in liquid or “soft” solid state, under stress, inparticular, when a mechanical force is exerted on the composition.Rheological properties can be measured from bulk sample deformationusing a mechanical rheometer, or on a micro-scale by using amicrocapillary viscometer or an optical technique such as microrheology.Examples of rheological properties include shear viscosity, elongationalviscosity, storage and loss moduli, viscoelastic properties, andlubrication properties.

Accordingly, in some embodiments herein described, associative polymersare provided which can be added to a non-polar composition to performdrag reduction, mist control, lubrication, fuel efficiency improvementand/or control of viscoelastic properties of a non-polar composition.

In particular, the term “drag reduction” as used herein refers to thereduction of the resistance to flow in turbulent flow of a fluid in aconduit (e.g. a pipe) or pipeline thereby allowing the fluid to flowmore efficiently. A skilled person would realize that drag reduction canbe described in terms that include, for example, a reduction in thefriction factor at high Reynolds number (e.g. higher than 5000, between5000 and 25000 and higher than 25000), a reduction in the pressure droprequired to achieve a given volumetric flow rate, a reduction inhydraulic resistance, and an increase in flow rate without raisingoperating pressure. In particular, drag reduction can be measured bymethods identifiable to a skilled person, for example measurement of theflow rate of a fluid though a conduit and/or by measurement of thechange in pressure of a fluid flowing through a conduit.

In particular, the term “mist control” as used herein refers to thecontrol of the properties of a fluid mist. In particular, the propertiesthat can be controlled can include the sizes, and/or distribution ofsizes, of the droplets of fluid. In some embodiments, control of thesizes, and/or distribution of sizes, of the droplets can control theflammability of the mist of a fluid (e.g., to reduce the propagation ofa flame through the fuel mist in the event of an accident). In otherembodiments, control of the sizes, and/or distribution of sizes, of thedroplets can increase the deposition of a fluid on an intended surface(e.g., to reduce pesticide wasted by convection away from the field towhich it is being applied). In particular, mist control can be measuredby techniques identifiable to a skilled person, such as measurement ofthe sizes and size distribution of droplets when a fluid is converted toa mist.

In particular, the term “lubrication” as used herein refers to thereduction of wear and/or inhibition of movement between two surfacesseparated by a non-polar composition as herein described. In particular,in some embodiments, the lubrication properties of a non-polarcomposition can be controlled to improve the wear-resistance and/ormovement of the surfaces with respect to each other when the non-polarcomposition is introduced as a lubricant between the two surfaces (e.g.improving the wear-resistance and/or movement of ball bearings in a ballbearing structure, or improving the wear resistance and/or movement of apiston in an engine). In particular, lubrication of a fluid can bemeasured by techniques identifiable to a skilled person, such asrheological measurements (e.g. measuring the coefficient of frictionwhen two surfaces with the fluid between them are slid past each other).

In particular, the term “fuel efficiency” as used herein, refers to thethermal efficiency with which the potential energy of a fuel isconverted to kinetic energy and/or work in the chemical transformationundergone by the fuel (e.g. combustion of the fuel in an engine). Inparticular, fuel efficiency can be measured by techniques identifiableto a skilled person, such as measurement of the amount of work performedby the chemical transformation of the fuel (e.g. measuring the number ofmiles of travel an engine can provide when combusting a given volume offuel).

In particular, the term “viscoelastic properties” as used herein refersto the manner in which a non-polar composition reacts to externalstresses such as deformation, in which the non-polar fluid exhibits acombination of viscous response (e.g. production of a permanent strainof the non-polar composition once it has been distorted by the appliedstress) and elastic response (deformation of the non-polar compositionduring application of the stress, and return to the original shape uponremoval of the stress). In particular, viscoelastic properties can bemeasured by methods identifiable to a skilled person, such asrheological measurements (e.g. measurement of the storage and lossmoduli of the non-polar composition).

Associative polymers herein described have a non-polar backbone andfunctional groups presented at ends of the non-polar backbone. Inparticular in the associative polymer the linear or branched backbone issubstantially soluble in the non-polar composition and in particular ina host composition. The term “substantially soluble” as used herein withreference to a polymer and a nonpolar composition indicates the abilityof the polymer backbone to dissolve in the non-polar liquid.Accordingly, the backbone of the associative polymers as hereindescribed can be substantially soluble in a nonpolar composition whenthe polymer backbone and nonpolar composition have similar Hildebrandsolubility parameters (6) which is the square root of the cohesiveenergy density:

$\delta = \sqrt{\frac{{\Delta H}_{v} - {RT}}{V_{m}}}$

wherein ΔH_(v) is equal to the heat of vaporization, R is the ideal gasconstant, T is the temperature, and V_(m) is the molar volume. Inparticular, similar solubility parameters between a polymer and anonpolar composition can be found when the absolute value of thedifference between their solubility parameters is less than about 1(cal/cm³)^(1/2) (see also Tables 3-5 herein). A skilled person willrealize that the ability of the backbone to dissolve in the non-polarcomposition can be verified, for example, by placing an amount of thehomopolymer or copolymer to be used as the backbone of the associativepolymer in a host liquid as herein described, and observing whether ornot it dissolves under appropriate conditions of temperature andagitation that are identifiable to a skilled person.

In some embodiments, the backbone of associative polymers as hereindescribed can be substantially soluble in a nonpolar composition whenthe difference in solubility parameters gives rise to a Flory-Hugginsinteraction parameter (χ) of about 0.5 or less. In particular, χ can bedetermined by the following empirical relationship:

$\chi = {{\chi_{s} + \chi_{H}} \approx {{{0.3}4} + {\frac{v_{0}}{RT}\left( {\delta_{1} - \delta_{2}} \right)^{2}}}}$

where χ_(s) is the entropic part of the interaction between theassociative polymer and nonpolar composition (generally assigned anempirical value of 0.34, as would be apparent to a skilled person),χ_(H) is the enthalpic part of the interaction, v₀ is the molar volumeof the nonpolar composition, δ₁ is the solubility parameter of thepolymer, and δ₂ is the solubility parameter of the host. Additionalexemplary empirical solubility parameters are identifiable by a skilledperson (see, e.g., [18] and other available references known oridentifiable by one skilled in the art) An exemplary solubilitydetermination of the backbone of an associative polymer according to thedisclosure with an exemplary non-polar composition is reported inExample 12. Similarly, a skilled person can determine if otherassociative polymer backbones would be substantially soluble in othernon-polar compositions by applying the same calculations using theparticular solubility parameters for the particular non-polarcomposition.

In embodiments herein described, associative polymers are polymershaving a non-polar backbone and functional groups presented at ends ofthe non-polar backbone and in particular at two or more ends of thenon-polar backbone.

The term “functional group” as used herein indicates specific groups ofatoms within a molecular structure that are responsible for thecharacteristic physical and/or chemical reactions of that structure andin particular to physical and/or chemical associative interactions ofthat structure. As used herein, the term “corresponding functionalgroup” or “complementary functional group” refers to a functional groupthat can react, and in particular physically or chemically associate, toanother functional group. Thus, functional groups that can react, and inparticular physically or chemically associate, with each other can bereferred to as corresponding functional groups. In some embodimentsherein described functional end groups of polymers to be added to a samenon-polar compositions are corresponding functional groups in the senseof the present disclosure.

In particular, exemplary functional groups can include such groups ascarboxylic acids, amines, and alcohols, and also molecules such as, forexample, diacetamidopyridine, thymine, the Hamilton Receptor (see, e.g.[19]), cyanuric acid, and others identifiable to a skilled person. Inparticular, some of the exemplary functional groups can form pairs ofcomplementary functional groups, for example, carboxylic acids withother carboxylic acids, carboxylic acids with amines, alcohols withamines, alcohols with carboxylic acids, diacetamidopyridine withthymine, the Hamilton Receptor with cyanuric acid, and othersidentifiable to a skilled person (see, e.g., FIG. 4).

In particular, in some embodiments, functional groups as hereindescribed can be synthesized by installation of other functional groupsonto the backbone of the associative polymers at a plurality ofappropriate ends as herein described and transformed according tomethods identifiable to a skilled person (see, e.g. [20]). Inparticular, in some of those embodiments the installation can beperformed in at least two ends of the associative polymers. Moreparticularly, installation at an end of the polymer can be performed byinstallation of the functional group on the terminal monomer of thepolymer backbone, or on an internal monomer within a range ofapproximately 1 to 100 monomers from the terminal monomer.

In associative polymer herein described, a number of the functionalgroups presented on ends of the backbone is formed by “associativefunctional groups” (herein also indicated as FGas) which are functionalgroup able to associate with each other and/or with correspondingfunctional groups in other associative polymers in a same non-polarcomposition with an association constant (k) in a range 0.1<log₁₀ k<18(preferably 2<log₁₀ k<18), so that the strength of each associativeinteraction is less than that of a covalent bond between backbone atoms.

In particular in associative polymer herein described associativefunctional groups are capable of undergoing an associative interactionone with another with an association constant (k)

${k\left( M^{- 1} \right)} \geq {\frac{\frac{4}{3}{\pi\left( R_{g}^{2} \right)}^{\frac{3}{2}}N_{a}}{n_{F}} \times 10^{{- 2}3}}$

in which R_(g) is the value of the radius of gyration of the associativepolymer in the non-polar composition in nanometers, N_(a) is Avogadro'sconstant; and n_(F) is the average number of associative functionalgroups per polymer molecule in the associative polymer.

In some embodiments, associative polymers can further comprisederivatizable functional group (herein also indicated as FGd) presentedat one or more ends of the at least two ends of the backbone. The term“derivatizable functional groups” refers to functional groups thatcannot form associative interactions one with another or with anassociative functional group in the non-polar composition and canundergo a derivatization reaction. The term “derivatization” is commonlyreferred to a technique in chemistry that transforms a chemical compoundinto a product of similar chemical structure, also called a derivative.A derivatizable functional groups refer to a specific type of functionalgroups that participate in the derivatization reaction and transform apolymer to its derivative having different chemical and/or physicalproperties such as reactivity, solubility, boiling point, melting point,aggregate state or chemical composition. Derivatizable functional groupscan be used in attach additional functional moieties (e.g. polydrugs seeExample 73) of the polymer of interest. Exemplary derivatizablefunctional groups FGd suitable for the associative polymers describedherein are typically non-polar FG that do not participate in hydrogenbonding and/or metal ligand coordination interactions, and possiblyallow coupling of functional moieties to the polymer. Exemplaryderivatizable functional groups comprise an azido group, an alkynylgroup, a thiol group, a vinyl group, a maleimide group, and additionalgroups identifiable by a skilled person (see e.g. FIG. 20 and FIG. 21)

In particular, in some embodiments, the at least two ends of theassociative polymers herein described presenting an associativefunctional group in the sense of the disclosure, identify at least twopositions in the linear, branched or hyperbranched polymer backbone ofthe associative polymer that are separated by an internal span that hasa length of at least 2,000 backbone bonds, or an internal span betweenfunctional groups with a weight average molar mass not less than 100,000g/mol. In embodiments herein described installation is performed so thatthe functional groups are presented on the polymer.

The terms “present” and “presented” as used herein with reference to acompound or functional group indicates attachment performed to maintainthe chemical reactivity of the compound or functional group as attached.The term “attach” or “attached” as used herein, refers to connecting oruniting by a bond, link, force or tie in order to keep two or morecomponents together, which encompasses either direct or indirectattachment where, for example, a first molecule is directly bound to asecond molecule or material, or one or more intermediate molecules aredisposed between the first molecule and the second molecule or material.

In particular, groups presented “at an end” of the polymer backbone cancomprise groups attached to a terminal monomer of a polymer or to amonomer less than 100 monomers from a terminal monomer of the polymerbased on the specific structure and configuration of the polymer as willbe understood by a skilled person upon reading of the presentdisclosure.

In various embodiments, functional end groups of associative polymersherein described are able to associate in a donor/acceptor associationand/or in a self-association (FIG. 1 and FIG. 2). In the donor/acceptorassociation the donor and acceptor can be stoichiometric (e.g. equalnumbers of donor and acceptor functional groups) or non-stoichiometric(e.g. more donor groups than acceptor groups or vice versa).

In various embodiments, the self-associative polymers, the backbone canbe linear or branched and following association of the associativefunctional end groups the self-associating polymer can form varioussupramolecular architectures (see Example 1). In particular in someembodiments the backbone length can be such that the backbone has aweight-average molecular weight of 250,000 g/mol and more for individualchains.

More particularly, in various embodiments, the backbone can be anonpolar linear, branched or hyperbranched polymer or copolymer (e.g.substituted or unsubstituted polydienes such as poly(butadiene) (PB) andpoly(isoprene), and substituted or unsubstituted polyolefins such aspolyisobutylene (PIB) and ethylene-butene copolymers, poly(norbornene),poly(octene), polystyrene (PS), poly(siloxanes), polyacrylates withalkyl side chains, polyesters, and/or polyurethanes) providing a numberof flexible repeat units between associative functional end groups.

In some embodiments, the weight-average molar mass (M_(w)) of theassociative polymer can be equal to or lower than about 2,000,000 g/moland in particular can be between about 100,000 g/mol and about 1,000,000g/mol.

In particular, in some embodiments, the backbone and associativefunctional end groups can be selected to have a ratio of carbon atoms toheteroatoms greater than about 1000:1 in the associative polymers. Forexample, in some embodiments, a skilled person can ensure that theheteroatom content is so low (e.g. greater than 10,000:1) as to notaffect burning (e.g. the emissions produced by burning a fuelcomposition that contains some associative polymers). In someembodiments, the associative polymer can comprise functional groupswithin the backbone as shown schematically in FIG. 6 and, therefore, ina location not limited to the functional groups at one or more end ofthe polymer backbone while still maintaining a ratio of carbon atoms toheteroatoms greater than about 1000:1.

In some embodiments associative polymers herein described and indicatedas framing associative polymer, comprise associative functional groupspresented at two or more ends of at least two ends of the backbone. Insome embodiments associative polymers herein described and indicated ascapping associative polymer, comprise an associative functional grouppresented at one end of the at least two ends of the backbone.

In embodiments herein described, the framing associative polymer can beused to control physical and/or chemical properties and in particularrheological properties of a non-polar composition alone or incombination with up to about 20% capping associative polymers. Inparticular in embodiments where capping associative polymers arecombined with framing associative polymers, the ability of the framingassociative polymers to control the properties of a non-polarcomposition is improved with respect to a comparable compositioncomprising framing associative polymers only (e.g. a 10% improved dragreduction). In some of those embodiments, the use of capping associativepolymers in combination with framing associative polymers allows use ofa reduced amount of framing associative polymers (e.g. 10%)

In embodiments herein described framing associative polymer and cappingassociative polymer can be linear, branched or hyperbranched polymerswith various structures as will be understood by a skilled person.

In associative polymers herein described, and in particular framingassociative polymer and capping associative polymer, the backbone of thepolymer can be characterized by a longest span.

A “longest span” in the sense of the disclosure is the greatest numberof backbone bonds between terminal monomers of the polymer among anypossible pairs of terminal monomers within the polymer. The longest spancan be measured base on the Radius of gyration of the polymer asdescribed for example [1{Iwao, 2002 #5452}] as will be understood by askilled person

A longest span of an associative polymer affects overall resistance ofan associative non-polar composition to elongational deformation. Inparticular, such resistance is dictated by the overall size of theassociative polymers herein described after association to formsupramolecules within the associative non-polar composition.Accordingly, in order to provide the greatest resistance to elongationdeformation of the associative non-polar composition, the associativepolymer can be selected to have the greatest possible longest span. Inembodiments, herein described where the associative non-polarcomposition is in a flow, however, hydrodynamic forces will be appliedto the associative polymers when the associative polymers are comprisedin a composition in a flow. In particular, the more turbulent the flowis, the greater the forces are that are applied to the associativepolymer within the composition. Depending on the extent of the forcesapplication a associative polymer in the composition can be stretched toits physical limit.

If the forces applied to a polymer exceed the maximum strength of thebackbone bond the backbone of the associative polymer, the associativepolymer will break typically in the middle section of the longest span.

In embodiments, herein described associative polymers are selected tohave a longest span having a contour length L, such that ½L_(b)≤L<L_(b)which is the length at which the longest span of the associative polymerwill not break when comprised in an associative non-polar composition ina flow characterized by set flow conditions.

A “contour length” in the sense of the disclosure indicates the lengthof a polymer when fully stretched along the longest span. In particular,a contour length can be expressed in nanometers. The contour length isdirectly proportional to the number of chemical bonds in the longestspan and therefore to the molecular weight of the longest span of theassociative polymer, as will be understood by a skilled person. Forexample, in a carbon based homopolymer the contour length of the longestspan L is L=n_(s) (0.82) (0.154 nm) wherein n_(s) is (M_(ws)/M₀)×n₀ inwhich M_(ws) is the M_(w) of the longest span, M₀ is the molecularweight of a repeating unit and no is the number of backbone chemicalbonds in the repeating unit.

L_(b) in the sense of the disclosure indicates a rupture length of theassociative polymer in nanometers when the associative polymer is withina host non-polar composition having a framing associative polymerconcentration c to provide an associative non-polar composition in aflow, L_(b) being given by implicit function

$F_{b} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{b} \right)}^{2}}{4{pd}^{2}{\ln\left( L_{b} \right)}} \times 10^{- 9}}$

in which F_(b) is the rupture force of the associative polymer innanonewtons, Re is the Reynolds number, d is the characteristic lengthof the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.

An “implicit function” is a mathematical equation that specifies adependent variable in terms of independent variables and parameters inwhich the dependent variable is not isolated on one side of theequation. Explicit functions give the dependent variable in terms of theindependent variables and parameters: if a dependent variable y can beisolated and equated to a function of an independent variable x, it isdescribed by an explicit function of the form y=f(x). In contrast, animplicit function is an equation that relates the dependent variable yto an independent variable x, which may not be solvable for y. Thesolutions to this equation are a set of points {(x,y)} which implicitlydefine a relation between x and y which is called an implicit function.The values of the function can be determined graphically using agraphing calculator or using a symbolic mathematical analysis program,such as Mathematica or Maple.

In associative polymers herein described F_(b) is the rupture force ofthe associative polymer is the rupture force of the associative polymer.F_(b) is a measure of the force required to break a polymer backbone andthe related value depends on the backbone structure as will beunderstood by a skilled person. In particular, a skilled person willunderstand that the weakest backbone bond usually determines the forcerequired to break the backbone as a whole. F_(b) can be measure byAtomic Force Microscopy (AFM) [{Grandbois, 1999 #5450}] density functiontheory calculation [{Schmidt, 2008 #5451}1], and other methodsidentifiable by a skilled person

In particular in embodiments herein described, F_(b) of associativepolymer herein described is preferably equal to or higher than 4.0 nN,and more particular equal to or higher than 4.1 nN.

A skilled person will be able to select a polymer backbone for theassociative polymer to be used in a non-polar composition under set flowconditions upon reading of the disclosure based on the state ofsubstitution of backbone atoms in polymers having a known F_(b). Forexample, the F_(b) value of chemical bonds such as Si—C (F_(b)=2.2 nN),Si—O (F_(b)=3.3 nN), and C—C (F_(b)=4.1 nN), are known or identifiableby a skilled person. Additionally, the rank ordering of bonddissociation energy will provide additional guidance in selection ofbackbone atoms as the rank ordering of bond dissociation energy tends tofollow trends in dissociation energy identifiable by a skilled person.For example, an entirely carbon backbone that contains double bonds andsingle bonds is expected to break at a single bond (C═C double bondaverage enthalpy 614 kJ/mol is much greater than that for a C—C singlebond, 348 kJ/mol).

A skilled person will be able to select a M_(w) of the longest span(M_(ws)) of an associative polymer based on the contour length L of theassociative polymer in nanometers based on the equation

$M_{ws} = {\left( \frac{M_{0}}{n_{0}} \right) \cdot \frac{L}{{\sin\left( \frac{{bond}\mspace{11mu}{angle}}{2} \right)} \cdot \left( {{bond}\mspace{11mu}{length}} \right)}}$

M₀ is the molecular weight of the repeating unit of the polymer, n₀ isthe number of backbone bond per repeating unit, bond angle indicates theaverage angle of the bonds in the fully stretched backbone of theassociative polymer, and bond length is the average length of the bondsin the fully stretched backbone of the associative polymer innanometers. A skilled person will be able to identify the bond angle andthe bond length in view of the type of backbone selected (e.g. in viewof a value of F_(b))

In embodiments herein described, a skilled person can calculate L_(b)based on the values of Re, F_(b), μ, ρ and d for the specificassociative non-polar composition and for the specific flow conditions,using the implicit equation herein described. In particular, L_(b) canbe the rupture length of the longest span of a framing associativepolymer (L_(bf)) and can be used to determine the contour length of thelongest span of a framing associative polymer (L_(f)), or can be therupture length of of the longest span of a capping associative polymer(L_(bc)), and can be used to determine the contour length of the longestspan of a capping associative polymer (L_(c)).

In embodiments herein described, calculation of L_(b), L_(bf), and/orL_(bc), is performed in function of the concentration c of framingassociative polymers in the associative non-polymer composition. Inparticular, in embodiments, where the concentration of framingassociative polymer c in the associative non-polar composition is c≤2c*,μ is the viscosity of the host non-polar composition μ_(h) and ρ is thedensity of the host non-polar composition ρ_(h). In embodiments wherethe concentration of framing associative polymer c in the associativenon-polar composition is c>2c*, μ is the viscosity of the associativenon-polar composition μ_(a), and ρ is the density of the associativenon-polar composition ρ_(a).

In embodiments herein described, the non-polar backbone of theassociative polymer presents functional groups at ends of the non-polarbackbone and in particular at two or more ends of the non-polarbackbone.

In particular embodiments, associative polymers herein described cancomprise one or more structural units of formula [[FG-chain

-[node]z-]-(I) and optionally the structural unit of formula -

node

-

chain]

- (II)

wherein:

-   -   FG is a functional group, which can comprise an associative        functional group FGa with one or more associative moieties such        that the functional group are capable of undergoing an        associative interaction with each other with an the association        constant (k) in a range 0.1<log₁₀ k<18 (preferably 2<log₁₀        k<18), so that the strength of each associative interaction is        less than that of a covalent bond between backbone atoms, or FG        can comprise a derivatizable functional group FGd with one or        more moieties capable of undergoing derivatization;    -   chain is a non-polar polymer substantially soluble in a        non-polar composition, the polymer having formula:

R₁-[A]_(n)R₂  (III)

-   -   wherein:        -   A is a chemical and in particular an organic or silicone            moiety forming the monomers of the polymer;        -   R₁ and R₂ are independently selected from any carbon or            silicon based or organic group with one of R₁ and R₂ linked            to an FG or a node and the other one of R₁ and R₂ linked to            an FG or a node; and        -   n is an integer ≥1;        -   z is 0 or 1, depending on the nature of the chemical link            between a unit of Formula (I) or Formula (II) and one or            more units of Formula (I) and/or Formula (II),        -   node is a covalently linked moiety linking one of R₁ and R₂            of at least one first chain with one of the R₁ and R₂ of at            least one second chain;    -   and wherein    -   the FG, chain and node of different structural units of the        polymer can be the same or different.

In embodiments herein described, din at least one structure structureunit having formula [[FG-chain

-[node]_(z)

- (I) and optionally in one or more structural units having formula -

node

-

chain]

- (II), n is ≥250 and in particular 300.

In particular, in associative polymers herein described includingstructural units of formula (I), FG groups presented “at an end” of thepolymer backbone can comprise groups attached to either a terminalmonomer of the chain or to a monomer less than 5% and possibly less than1% of the total number of monomers of the chain from the terminalmonomer of the chain in a structural unit of Formula I).

Associative polymers and in particular framing associative polymers andcapping associative polymers in accordance with the present disclosurecan comprise one or more of the structural units according to Formula(I) and/or Formula (II) in various configurations as would be apparentto a skilled person upon reading of the present disclosure.

For example in some embodiments herein described framing associativepolymers comprise at least two structural units of Formula (I) whereinFG is an FGa. In some embodiments, framing polymers herein described cancomprise additional structural units of Formula (I) and/or Formula (II)possibly presenting additional FGas.

In some embodiments herein described, capping associative polymerscomprise one structural unit of Formula (I) wherein FG is an FGa. Insome embodiments, the capping associative polymers can compriseadditional structural units of Formula (II).

In some embodiments, framing associative polymers herein described canbe formed by three or more structural units of Formula (I), wherein atleast two of the structural units of Formula (I) are attached one toanother with a structural unit of formula -

node

-

chain]

- (II) and wherein each [node] attaches three structural unit of Formula(I). In some of those embodiments, all the FGs are FGas. In some ofthose embodiments, structural unites of Formula (I) can be distancedfrom one another.

In some embodiments, the framing associative polymer can be formed bytwo structural units of Formula (I) wherein in the first structural unitz is 0 and in the second structural unit z is 1 and the node of thesecond structural unit links to one of R₁ and R₂ of the first structuralunit thus forming a linear polymer. In some of those embodiments, theassociative polymer is a framing associative polymer and the FGs areFGas.

In polymers comprising structural units of Formula (I) and, optionally,structural units of Formula (II), the longest span of the polymer is thegreatest number of backbone bonds between terminal monomers of thepolymer comprising the structure units Formula (I) and optionallystructural units of Formula (II) among any possible pairs of terminalmonomers within the polymer. A longest span can have the formFG-chain-node-chain-FG in the case of polymers that contain onlystructural units of Formula (I), or can have the formFG-chain-node-[chain-node]n-chain-FG for chains that include both typesof units. Knowledge of the mean value of the length of the -chain-unitscan be used to estimate the average length of the longest span.

The longest span controls rupture of the polymer when the polymer is ina non-polar composition subjected to a flow, and in particular aturbulent flow as will be understood by a skilled person [{Church, 2014#5453}].

In the synthesis of branched polymers, the method of synthesis oftencontrols the type and extent of branching. In the case of polymers thatcontain only structural units of Formula (I), the architecture is eitherlinear or star-type. If the average degree of polymerization of the-chain-units is N_(c), the average span of the polymer is 2N_(c) forlinear and star polymers having a modest number of arms (e.g, 6 arms, oranother number that results in no crowding). For a polymer that hasfirst generation branches only (H-shaped or comb-shaped polymers), thelongest span is simply related to the number of structural units ofFormula (II) that separate the structural units of Formula (I) at eachend of the H- or comb-shaped polymer. For example, an H-shaped polymerhas an average number of monomer units in the longest span that is3N_(c). If branch-on-branch structure is present, similar reasoningholds. For example, if the polymer has two generations of tri-functionalbranching, the longest span contains, on average, 4N_(c) repeat units.

An estimate of the number of monomer units in the longest span can beused to estimate the radius of gyration that a branched polymer willhave, because R_(g) of lightly branched polymers is only slightlygreater than it would be for a linear chain of the same length as thelongest span.

In many applications of the associative polymers herein described, thepolymer backbone is selected such that it dissolves substantially wellin the host of interest. Therefore, good solvent conditions usuallyprevail. Using the scaling relationships for good solvent and theestimated degree of polymerization, useful estimates of the radius ofgyration can be calculated. In turn, these can be used in preliminarydesign calculations. Such preliminary calculations can guide theselection of molecules to synthesize. Once the polymers have beenprepared, the value of R_(g) can simply be measured using such methodsas static light scattering or viscometry.

In some embodiments herein described, FG indicates a functional groupFGa that is capable of undergoing an associative interaction withanother suitable functional group whereby the association constant (k)for an interaction between associating functional groups is in the range0.1<log₁₀ k<18, and in particular in the range 4<log₁₀ k<14 so that thestrength of each individual interaction is less than that of a covalentbond between backbone atoms. In particular, in some embodiments, the FGacan be chosen to have an association constant that is suitable for agiven concentration of the associative polymer in the non-polarcomposition relative c*, as described herein. For example, a skilledperson will realize that if the concentration of the associative polymeris high (e.g. greater than 3c*), a lower log₁₀ k value (e.g. about 4 toabout 6) can be suitable, as can a higher log₁₀ k value (e.g. about 6 toabout 14). Additionally, a skilled person will also realize that if theconcentration of associative polymer is low (e.g. less than 0.5c*) ahigher log₁₀ k value (e.g. about 6 to about 14) can be suitable.

Exemplary FGaS comprise those that can associate through homonuclearhydrogen-bonding (e.g. carboxylic acids, alcohols), heteronuclearhydrogen-bonding donor-acceptor pairing (e.g. carboxylic acids-amines),Lewis-type acid-base pairing (e.g. transition metal center-electron pairdonor ligand such as palladium (II) and pyridine, or iron andtetraaceticacid, or others identifiable to a skilled person as moietiesthat participate in metal-ligand interactions or metal-chelateinteractions), electrostatic interactions between charged species (e.g.tetraalkylammonium-tetraalkylborate), pi-acid/pi-base or quadrupoleinteractions (e.g. arene-perfluoroarene), charge-transfer complexformation (e.g. carbazole-nitroarene), and combinations of theseinteractions (e.g. proteins, biotin-avidin). More than one type of FGsand in particular of FGas may be present in a given polymer structure.

In some embodiments, FGa can be selected among a diacetamidopyridinegroup, thymine group, Hamilton Receptor group (see, e.g. [19]), cyanuricacid group, carboxylic acid group, primary secondary or tertiary aminegroup, primary secondary and tertiary alcohol group, and othersidentifiable to a skilled person.

In some embodiments, in the structural unit of Formula (I), FG can be aderivatizable functional group (FGd). Exemplary derivatizable FGdscomprise of an azido group, an alkynyl group, a thiol group, a vinylgroup, a maleimide group, and additional groups identifiable by askilled person (see e.g. FIGS. 20 and 21).

In the structural unit of Formulas (I) and (II) a chain can be a polymerbackbone that is substantially soluble in a liquid host that has adielectric constant equal to or less than 5. Such chains can comprisefor example polydienes such as poly(butadiene), poly(isoprene),polyolefins such as polyisobutlyene, polyethylene, polypropylene andpolymers of other alpha olefins identifiable to a skilled person,poly(styrene), poly(acrylonitrile), poly(vinyl acetate),poly(siloxanes), substituted derivatives thereof, and copolymers ofthese.

In the structural unit of Formulas (I) and (II) a node can be aconnecting unit between one or more and in particular two or more[FG-chain] units such that the total molecular structure issubstantially terminated by FG species (e.g., a plurality of the chainends have a FG less than 100 repeat units from the chain end). In someembodiments, the simplest such polymer is a linear telechelic: two[FG-chain] units with their chains connected end-to-end at a node:[FG-chain]-node-[chain-FG] or FG-chain-FG. Alternative branched,hyperbranched, star, brush, partially-cross linked or other multi-armedpolymer structures can also be used, provided that ends and/or otherregions of the polymer chain are functionalized according to the presentdisclosure. In particular, a skilled person will understand from areading of the present disclosure the term “functionalized” according tothe present disclosure can be understood to mean that the functionalgroups can be at the end of the polymer chains or other polymerstructures, or at different regions within the polymer chain (see, e.g.,FIGS. 5 and 6).

In particular, in certain cases, the nodes can comprise one or more FGunits formed by FGa such that some degree of associative functionalityis present in the internal polymer structure. A node is formed by anycovalently bound group such as organic, siloxane, and additional groupidentifiable by a skilled person. In particular, a node can link two ormore chains through suitable covalent bonds and more particularly formbranched polymers wherein a node can link two to 10 chain -

node

-

chain]

- (II) (see e.g. FIG. 5). More than one type of nodes may be present ina given polymer structure. In some embodiments the node can be atertiary carbon, a cycloaliphatic moiety or an aliphatic chain.

In particular in some embodiments, the chain can have a formulaR₁[A]_(n)-R₂ (III) in which A is a chemical moiety suitable to be usedas monomer and n can indicate the degree of polymerization of the chain.In some embodiments, n can be an integer equal to or greater than 200and, in particular, equal to or greater than 800. In some embodiments Acan be an organic moiety having secondary carbon atoms, tertiary carbonatoms and/or quaternary carbon atoms, as will be understood by a skilledperson. In some of those embodiments A can be an organic moietycomprising up to 10% of tertiary carbon atoms.

In some embodiments particular A can be a diene, olefin, styrene,acrylonitrile, methyl methacrylate, vinyl acetate,dichlorodimethylsilane, tetrafluoroethylene, acids, esters, amides,amines, glycidyl ethers, isocyanates and additional monomersidentifiable by a skilled person. The term “olefins” as used hereinindicates two carbons covalently bound to one another that contain adouble bond (sp²-hybridized bond) between them. Olefins include alphaolefins and internal olefin.

E₁, E₂ and E₃ are selected independently from hydrogen and linear,branched or cyclic C1-C24 alkyl, preferably C1-C12 alkyl, morepreferably C1-C8 alkyl including methyl, ethyl, butyl, propyl, hexyl,and ethylhexyl.

In particular, a skilled person will realize that the particularmoieties used as monomers can give rise to polymer backbones that aresuitable for combination with particular types of nonpolar compositions.For example, styrene monomers, olefin monomers, and in particular dienemonomers can form polymers for very non-polar compositions (e.g.compositions with a dielectric constant of 1.5-2.5); amide, ester,epoxy, and urethanes can form polymers for nonpolar compositions thathave somewhat greater dielectric constants (e.g., in the range 2.5-5);and fluorocarbon monomers and silicone monomers can form polymers forfluorous media. A skilled person will understand that additional typesof monomers would be suitable for other types of nonpolar compositions.

In some embodiments, A in Formula (III) can be a moiety selected toprovide a chain of formula (IV):

wherein R^(a)-R^(m) are independently selected from hydrogen, C₁-C₁₂substituted or unsubstituted alkyl, cycloalkyl, alkeneyl, cycloalkenyl,alkynyl, cycloakynyl, and aryl groups and n is in the range 200-20,000and, in particular, in the range from 1000-10,000.

In some embodiments, A in formula (III) can be a moiety selected toprovide a chain of formulas (V)-(VIII):

wherein R^(a)-R^(j) are independently selected from the group consistingof hydrogen, C₁-C₁₂ substituted or unsubstituted alkyl, cycloalkyl,alkenyl, cycloalkenyl, alkynyl, cycloakynyl, and aryl groups and n is1000-20,000.

In some embodiments, A in formula (III) can be a moiety selected toprovide a chain of formula (IX):

wherein R^(a)-R^(d) are independently selected from the group consistingof hydrogen, C₁-C₁₂ substituted or unsubstituted alkyl, cycloalkyl,alkenyl, cycloalkenyl, alkynyl, cycloakynyl, and aryl groups and n is1000-40,000.

In some embodiments, A in formula (III) can be a moiety selected toprovide a chain of formula (X):

wherein R^(a)-R^(h) are independently selected from the group consistingof hydrogen, C₁-C₁₂ substituted or unsubstituted alkyl, cycloalkyl,alkenyl, cycloalkenyl, alkynyl, cycloakynyl, and aryl groups and n is1000-20,000.

In some embodiments, A in formula (III) can be a moiety selected toprovide a chain of formula (XI):

wherein R^(a)-R^(e) are independently selected from the group consistingof hydrogen, C₁-C₁₂ substituted or unsubstituted alkyl, cycloalkyl,alkenyl, cycloalkenyl, alkynyl, cycloakynyl, and aryl groups and n is1000-20,000.

In embodiments of the nodes of Formula (III) R₁ and R₂ can be chemicalmoieties independently selected and capable of forming a covalent bond.In some embodiments, either R₁ or R₂ of at least one first chain can belinked to one of the R₁ and R₂ of at least one second chain through anode. In some embodiments, a node can comprise functional groups such asarenes, perfluoroarenes, groups containing oxygen, groups containingnitrogen and groups containing phosphorus and sulfur all identifiable bya skilled person. In particular, functional groups suitable for nodescan comprise a carboxylic acid, amine, triarylphosphine, azide,acetylene, sulfonyl azide, thio acid and aldehyde. In particular, forexample, in forming covalent links between node and chain and possiblybetween node and functional group a first chemical moiety and a secondcorresponding chemical moiety can be selected to comprise the followingbinding partners: carboxylic acid group and amine group, sulfonyl azideand thio acid, and aldehyde and primary amine. Additional chemicalmoieties can be identified by a skilled person upon reading of thepresent disclosure. Reference is also made to the exemplary nodes ofExample 11.

In some embodiments, R₁ and/or R₂ can be R₁ and R₂ are independentlyselected from a divalent group or atom.

In some embodiments where A is a moiety selected to provide a chain offormula (IV)-(VIII), (X), or (XI), R₁ and/or R₂ can be a moiety offormula (XII):

wherein:q is 1 to 18;X is selected from the group consisting of CH₂, O, and S; andR^(a) and R^(b) are independently hydrogen and/or a moiety of formulaXIII-XVIII:

provided that at least one of R^(a) and/or R^(b) is not hydrogen. Inparticular R^(a) and R^(b) can be FGs connected to the chain through R₁or R₂ of Formula XII.

In some embodiments where A is a moiety selected to provide a chain offormula (IV)-(VIII), (X), or (XI), R₁ and/or R₂ can be a moiety offormula (XX):

wherein:q is 1 to 18;X is selected from the group consisting of CH₂, O, and S; andR^(a) and R^(b) are independently a moiety of formula (XIII)-(XVIII) asdescribed herein. In particular R^(a) and R^(b) can be FGs connected tothe chain through R₁ or R₂ of Formula (XX).

In some other embodiments where A is a moiety selected to provide achain of formula (IV)-(VIII), (X), or (XI), R₁ and/or R₂ can be a moietyof formula (XX):

wherein:q is 1 to 18;X¹, X², and X³ are independently selected from the group consisting ofCH₂, O, and S; andR^(a)-R^(d) are independently hydrogen and/or a moiety of formula(XIII)-(XVIII) as described herein; provided that at least one of R^(a),R^(d), R^(c), and/or R^(b) is not hydrogen. In particular R^(a), R^(b),R^(c) and R^(d) can be FGs connected to the chain through R₁ or R₂ ofFormula (XX).

In some other embodiments where A is a moiety selected to provide achain of formula (IV)-(VIII), (X), or (XI), R₁ and/or R₂ can be a moietyof formula (XXI):

wherein:q, r and s are independently 1 to 18;X¹, X², and X³ are independently selected from the group consisting ofCH₂, O, and S; andR^(a)-R^(d) are independently hydrogen and/or a moiety of formula(XIII)-(XVIII) as described herein; provided that at least one of R^(a),R^(b) R^(c), and/or R^(d) is not hydrogen. In particular R^(a), R^(b),R^(c) and R^(d) can be FGs connected to the chain through R₁ or R₂ ofFormula (XXI).

In some embodiments nodes can also present additional groups for bindingwith FG which can be introduced at the node according to someembodiments. In some embodiments nodes comprise an organic moiety, insome embodiments nodes comprise non organic moieties such as

Si—O

and additional moieties identifiable by a skilled person.

In some embodiments where A is a moiety selected to provide a chain offormula (IX) R₁ and/or R₂ can be a moiety of formula (XXII):

wherein:q is 1 to 18;X is selected from the group consisting of CH₂, O, and S; andR^(a) and R^(b) are independently H and/or a moiety of formula(XIII)-(XVIII) as described herein, provided that at least one of R^(a)and/or R^(b) is not H. In particular, R^(a) and R^(b) can be FGsconnected to the chain through R₁ or R₂ of Formula (XXII).

In some embodiments where A is a moiety selected to provide a chain offormula (IX) R₁ and/or R₂ can be a moiety of formula (XXIII):

wherein:q is 1 to 18;X is selected from the group consisting of CH₂, O, and S; andR^(a) and R^(b) are independently a moiety of formula (XIII)-(XVIII) asdescribed herein. In particular, R^(a) and R^(b) can be FGs connected tothe chain through R₁ or R₂ of Formula (XXIII).

In some other embodiments where A is a moiety selected to provide achain of formula (IX) R₁ and/or R₂ can be a moiety of formula (XXIV):

wherein:q is 1 to 18;X¹, X², and X³ are independently selected from the group consisting ofCH₂, O, and S; andR^(a)-R^(d) are independently H and/or a moiety of formula(XIII)-(XVIII) as described herein; provided that at least one of R^(a),R^(b), R^(c), and/or R^(d) is not H. In particular R^(a), R^(b), R^(c)and R^(d) can be FGs connected to the chain through R₁ or R₂ of Formula(XXIV).

In some other embodiments where A is a moiety selected to provide achain of formula (IX) R₁ and/or R₂ can be a moiety of formula (XXV):

wherein:q, r and s are independently 1 to 18;X¹, X², and X³ are independently selected from the group consisting ofCH₂, O, and S; andR^(a)-R^(d) are independently H and/or a moiety of formula(XIII)-(XVIII) as described herein; provided that at least one of R^(a),R^(b), R^(c), and/or R^(d) is not H. In particular R^(a), R^(b), R^(c)and R^(d) can be FGs connected to the chain through R₁ or R₂ of Formula(XXV).

In some other embodiments where A is a moiety selected to provide achain of formula (IX) R₁ and/or R₂ can be a moiety of formula (XXVI):

wherein:q is 1-18;R^(a)-R^(b) are independently H and/or a moiety of formula(XIII)-(XVIII) as described herein; and R^(c) is hydrogen or C₁-C₁₂substituted or unsubstituted alkyl; provided that at least one of R^(a),R^(b), and/or R^(c) is not H. In particular, R^(a), R^(b), and R^(c) canbe FGs connected to the chain through R₁ or R₂ of Formula (XXVI).

In some other embodiments where A is a moiety selected to provide achain of formula (IX) R₁ and/or R₂ can be a moiety of formula (XXVII):

wherein.q is 1 to 18;R^(a)-R^(d) are independently H and/or a moiety of formula(XIII)-(XVIII) as described herein; and R^(f)-R^(g) are independentlyhydrogen or C₁-C₁₂ substituted or unsubstituted alkyl; provided that atleast one of R^(a), R^(b), R^(c), and/or R^(d) is not H. In particular,R^(a), R^(b) R^(c) and R^(d) can be FGs connected to the chain throughR₁ or R₂ of Formula (XXVII).

In particular in some embodiments the [chain-node] segments have weightaverage molecular weight equal to or greater than 10,000 g/mol. In someembodiments the span of [chain-node]m between FGs has average molarmass >50,000 g/mol (in particular when dispersion in the hostcomposition despite the “solvent-phobic” FGas is desired). In someembodiments, the largest span of the molecule can be equal to or lessthan 500,000 g/mol (for example, when resistance to shear degradation isdesired). In some embodiments the largest span of the molecule,expressed as weight average molecular weight can be equal to or lessthan 1,000,000 g/mol.

In some embodiments, associative polymers herein described can betelechelic.

In some embodiments, associative polymers herein described have a totalpolymer molecular weight is M_(w)≤2,000,000 g/mol and in particular canbe between 100,000 g/mol and 1,000,000 g/mol. In some embodiments thelargest span between nodes is less than 500,000 g/mol in particular whenthe associative polymers are branched polymers.

In some embodiments, selection of molecular weight for an associativepolymer herein described can be performed in view of factors hereindescribed and in particular values of the binding constant in view ofavailable or desired FGas, and a desired concentration in view of effectto be controlled. Additional factors that can be considered comprise adesired viscosity of the host composition (e.g. high M_(w) at lowconcentration to minimize impact on the shear viscosity of the host andlower M_(w) at high concentration to increased impact on the shearviscosity of the host), a desired density of FGs and in particular FGaspresented in connection with a desired effect (e.g. in order to obtaingelation, concentrations near or greater than the overlap concentrationof the polymers are preferred), and duration of the control in view ofthe shear degradation (e.g. if a longer duration of the control isdesired, the longest span of the molecules can be reduced below thethreshold chain length for shear degradation in the application ofinterest)

In some embodiments, associative polymers herein described can have aweight-average molecular weight equal to or higher than about 100,000g/mol.

In some embodiments, associative polymers herein described can have aweight-average molecular weight between 400,000 to 1,000,000 g/mol.

In some embodiments, associative polymers herein described can have aweight-average molecular weight between 630,000 g/mol to 730,000 g/mol.

In some embodiments, associative polymers herein described can have aweight-average molecular weight between 100,000 g/mol to 300,000 g/mol.

In some embodiments, associative polymers herein described can have aweight-average molecular weight between 300,000 g/mol to 700,000 g/mol.

In some embodiments, associative polymers herein described can have aweight-average molecular weight between 700,000 g/mol to 1,000,000g/mol.

In some embodiments, associative polymers herein described can have aweight-average molecular weight between 1,000,000 g/mol to 2,000,000g/mol.

In some embodiments associative polymers herein described can have anatomic composition with heteroatoms (i.e., other than C or H) present atless than 1 heteroatom per 1000 carbons. In some embodiments,heteroatoms are placed predominantly in correspondence of the functionalgroups.

In some embodiments associative polymers herein described can have asignificant level of unsaturation (e.g. with a ratio of H to C less than1.8), which can improve low temperature liquid behavior. However,fully-saturated chains can also be considered effective and are includedin the scope of the current disclosure.

In various embodiments herein described, the associative polymers of thedisclosure can interact to form supramolecular structures followinginteractions of the FGa having association constant (k) of from0.1<log₁₀ k<18 and in particular from 6<log₁₀ k<14, in cases dragreduction and/or flow rate enhancement are desired.

In some embodiments, selection of binding constant for an associativepolymer herein described can be performed in view of factors hereindescribed and in particular values of M_(w) desired, available ordesired FGas, and a desired concentration in view of effect to becontrolled. Additional factors that can be considered comprise thespecific host composition in which the polymer is used, and additionalfactors identifiable by a skilled person upon reading of the presentdisclosure.

In some embodiments, associative polymers herein described can have anassociation constant 2≤log₁₀ k≤18.

In some embodiments, associative polymers herein described can have anassociation constant 4≤log₁₀ k≤14.

In some embodiments, associative polymers herein described can have anassociation constant 4≤log₁₀ k≤12.

In some embodiments, associative polymers herein described can have anassociation constant 6≤log₁₀ k≤14.

In some embodiments, associative polymers herein described can have anassociation constant 6.9≤log₁₀ k≤7.8.

In some embodiments, associative polymers herein described can have anassociation constant log₁₀ k≤14 in particular when the weight-averagemolecular weight equal to or lower than about 2,000,000 g/mol.

In some embodiments, associative polymers herein described can have anassociation constant 5.5≤log₁₀ k in particular when the weight averagemolecular weight equal to or higher than about 100,000 g/mol.)

In some embodiments, associative polymers herein described can have anassociation constant 7≤log₁₀ k≤9, in particular when the aweight-average molecular weight is between 400,000 to 1,000,000 g/mol.

In some embodiments, associative polymers herein described can have anassociation constant 6.9 log₁₀ k≤7.8 in particular when theweight-average molecular weight is between 630,000 g/mol to 730,000g/mol.

In some embodiments, associative polymers herein described can have anassociation constant 6≤log₁₀ k≤14, preferably 6≤log₁₀ k≤7.5, inparticular when the weight-average molecular weight is between 100,000g/mol to 300,000 g/mol.

In some embodiments, associative polymers herein described can have anassociation constant 6.9≤log₁₀ k≤14, preferably 6.9≤log₁₀ k≤7.8 inparticular when the weight-average molecular weight between 300,000g/mol to 700,000 g/mol.

In some embodiments, associative polymers herein described can have anassociation constant 7≤log₁₀ k≤14, and preferably 7≤log₁₀ k≤9 inparticular when the weight-average molecular weight between 700,000g/mol to 1,000,000 g/mol.

In some embodiments, associative polymers herein described can have anassociation constant 7.5≤log₁₀ k≤14, preferably 7.5≤log₁₀ k≤12, inparticular when the weight-average molecular weight between 1,000,000g/mol to 2,000,000 g/mol. In particular, in embodiments herein describedwhere drag reduction is desired in flows having a Reynolds numberbetween 5,000 and 25000, polymers and related FGs can be selected tohave an FGas with an association constant between 4≤log 10 k≤12, and inparticular 5.5≤log 10 k≤12 and in flows having a Reynolds number equalto or higher than 25,000 polymers and related FGs can be selected tohave an association constant between: 6≤log 10 k≤14.

In particular, in embodiments of supramolecular structures, FGaassociations can be due to, for example reversible noncovalentinteraction between the associative polymers that enables a discretenumber of molecular subunits or components to be assembled, typicallywith an individual interaction strength less than that of a covalentbond. Exemplary interactions include, for example, self-associativehydrogen bonds (H-bonds), donor-acceptor H-bonds, Brønsted or Lewisacid-base interactions, electrostatic interactions, pi-acid/pi-base orquadrupolar interactions, charge transfer complex formation, or othersupramolecular interactions.

In various embodiments herein described, the associative polymers of thepresent disclosure can be used in connection with a non-polarcomposition to control rheological properties, such as drag reductionand/or flow rate enhancement, sizes, and/or size and size distributionthe droplets of a fluid mist, and viscoelastic properties of thecomposition alone or in combination with other physical and/or chemicalproperties of the composition. In particular, in some embodiments, thenon-polar compositions comprise a host composition and at least oneframing associative polymer herein described.

The terms “host” and “host composition,” as used herein, refer to amajority component in a non-polar composition in which the physicaland/or chemical properties are sought to be controlled. In particular,the host or host composition can be a single substance such as a solventlike hexane or benzene, or the host or host composition can be asubstance which is a mixture such as gasoline, diesel, olive oil, orkerosene. The host or host composition can also be a mixture such as apaint or ink.

In some embodiments, the host composition can be a hydrocarboncomposition, a fluorocarbon compositions or a silicone composition, Insome embodiments, the host composition can be a biofuel, a mineral oil,crude oils, pentane, hexane, cyclohexane, benzene, toluene, chloroformand diethyl ether, dimethyl ether, liquefied petroleum gas, liquidmethane, butane, gasoline, kerosene, jet fuel and diesel fuel.

In particular, in non-polar compositions herein described a range ofhosts can have dielectric constant less than 5, with hosts havingdielectric constant less than 2.5 being particularly well suited toapplications herein described as will be understood by a skilled personupon reading of the disclosure. Non-polar compositions with the abovementioned dielectric constants encompasses a wide range of liquids thatare relevant to applications that comprise fuels (such as gasoline,kerosene, jet fuel, diesel and additional fuels identifiable by askilled person), foods and pharmaceuticals (such as olive oil, linseedoil, castor oil and additional foods identifiable by a skilled person),solvents used as cleaning fluids (such as turpentine, toluene andadditional solvents identifiable by a skilled person), and adhesiveformulations (such as pinene and additional formulations identifiable bya skilled person).

In embodiments of non-polar composition of the present disclosure, thedielectric constant of a given host will vary with temperature, whichcan be taken into account by one skilled in the art.

Exemplary non-polar compositions, and in particular host liquids, with adielectric constant less than 5 are illustrated in the table below(Table 1A). The table also provides exemplary hosts that can berecognized as unfavorable for the modified non-polar compositions hereindescribed (see Table 1B).

TABLE 1A Entry Fluid Temperature/° C. Dielectric constant ε ExemplaryFavorable Hosts  1   Benzene 20   2.3   2   Carbon disulfide 2.64  3  Carbon tetrachloride 20   2.23  4   Castor oil 15.6 4.7   5   Chloroform20   4.8   6   Cotton seed oil 3.1   7   Cumene 20   2.4   8   Decane20   2     9   Dodecane 20   2    10   Ether 20   4.3  11   Fluorinerefrigerant R-12 25   2    12   Fluorine refrigerant R-22 25   2    13  Furan 25   3    14   Gasoline 21.1 2    15   Heptane 20   1.9  16  Hexane −90    2    17   Jet fuel 21.1 1.7  18   Kerosene 21.1 1.8  19  Linoleic acid  0   2.6-2.9 20   Linseed oil 3.2-2.5 21   Naphthalene20   2.5  22   Octane 20   2    23   Olive oil 20   3.1  24   Palmiticacid 71.1 2.3  25   Pentane 20   1.8  26   Phenol 10   4.3  27   Pinene20   2.7  28   Styrene 25   2.4  29   Terpinene 21.1 2.7  30   Toluene2.0-2.4 31   Turpentine (wood) 20   2.2  32   Vacuum (by definition)1    32.1 Cyclohexane 2.0  32.2 Liquid methane −280     1.7  32.3 LiquidButane −1   1.4  32.4 Heavy oil 3    32.5 Petroleum oil 2.1  32.6 Liquidasphalt 2.5-3.2

TABLE 1B Entry Fluid Temperature/° C. Dielectric constant ε ExemplaryFavorable Hosts 33 Acetone 25   20.7 34 Alcohol, ethyl 25   24.3(ethanol) 35 Alcohol, methyl 20   35.1 (methanol) 36 Alcohol, propyl20   21.8 37 Ammonia (aqua) 20   15.5 38 Aniline 20    7.3 39 Cresol17.2 10.6 40 Ethylamine 21.1  6.3 41 Ethylene glycol 20   37   42Furfural 20   42   43 Glycerine  47.68 44 Glycerol 25   42.5 45 Hexanol25   13.3 46 Hydrazine 20   52   47 Pyridine 20   12  

In particular, in some embodiments, host composition that havedielectric constant equal to or less than about 5 are pentane, hexane,cyclohexane, benzene, toluene, chloroform and diethylether. In someembodiments, which can be used for fuel applications host compositioncan also have dielectric constant less than 5, including liquifiedpetroleum gas, liquid methane, butane, gasoline, kerosene, jet fuel anddiesel fuel.

In embodiments, herein described polymer dielectric constants canfurther provide an indication of their compatibility with a chosennon-polar composition that is in the range indicated in above. Referenceis made for example to the exemplary list provided in the table below(Table 2).

TABLE 2 Dielectric Constant Plastic Material -ε- Acetal 3.7-3.9 Acrylic2.1-3.9 ABS* 2.9-3.4 Polybutadiene approximately 2 Polycarbonate 2.9-3.8Polyester, TP 3.0-4.5 Polypropylene 2.3-2.9 Polysulfone 2.7-3.8Polydimethylsiloxane 3.0-3.2 (Silicone Rubber) Polyphenylene sulfide2.9-4.5 Polyacrylate 2.6-3.1 *ABS is Acrylonitrile Butadiene Rubber

In particular, in some embodiments, for a given host determined to havea dielectric constant within the threshold herein disclosed, at leastone framing associative polymer and optionally one or more cappingassociative polymers herein described are selected that aresubstantially soluble in the host in accordance with the presentdisclosure.

In particular, appropriate associative polymers for a given host can beidentified by a skilled person in view of the present disclosure. Forexample the backbone substantially soluble in the host composition canbe identified by comparison of the solubility parameters (6) of thepolymer backbone and host composition, as well as by determining theFlory-Huggins interaction parameter (χ) from the solubility parametersaccording to calculations described herein. In an exemplary embodiment,one or more polymer-solvent pairs can have silicone backbones for use inone or more fluorocarbon liquids.

In particular, an exemplary reference providing solubility parameters isthe websitewww.sigmaaldrich.com/etc/medialib/docs/Aldrich/General_Information/polymer_solutions.Par.0001.File.tmp/polymer_solutions.pdfat the time of filing of the present disclosure (see Tables 3-5). Moreparticularly, a skilled person will know that Sigma-Aldrich and otherchemical companies provide exemplary tables showing exemplary solubilityparameter values for various non-polar compositions and polymers. Askilled person can also refer to sources such as the Polymer Handbook tofind solubility parameter values [18].

TABLE 3 Table II: Solubility Parameters for Plasticizers and Solvents(Alphabetical sequence)

H-Bonding Solvent

Acetone 9.9 m Acetonitrile 11.9  p Amyl acetate 8.5 m Aniline 10.3  sBenzene 9.2 p Butyl acetate 8.3 m Butyl alcohol 11.4  s Butyl butyrate8.1 m Carbon disulfide 10.0  p Carbon tetrachloride 8.6 p Chlorobenzene9.5 p Chloroform 9.3 p Cresol 10.2  s Cyclohexanol 11.4  s Diamyl ether7.3 m Diamyl phthalate 9.1 m Dibenzyl ether 9.4 m Dibutyl phthalate 9.3m Dibutyl sebacate 9.2 m 1,2-Dichlorobenzene 10.0  p Diethyl carbonate8.8 m Di(ethylene glycol) 12.1  s Di(ethylene glycol) monobutyl 9.5 mether (Butyl Carbitol 

) Di(ethylene glycol) monoethyl 10.2  m ether (Carbitol 

) Diethyl ether 7.4 m Diethyl ketone 8.8 m Diethyl phthalate 10.0  mDi-n-hexyl phthalate 8.9 m Diisodecyl phthalate 7.2 mN,N-Dimethylacetamide 10.8  m Dimethyl ether 8.8 m N,N-Dimethylformamide12.1  m Dimethyl phthalate 10.7  m Dimethylsiloxanes 4.9-5.9 p Dimethylsulfoxide 12.0  m Dioctyl adipate 8.7 m Dioctyl phthalate 7.9 m Dioctylsebacate 8.6 m 1,4-Dioxane 10.0  m Di(propylene glycol) 10.0  sDi(propylene glycol) 9.3 m monomethyl ether Dipropyl phthalate 9.7 mEthyl acetate 9.1 m Ethyl amyl ketone 8.2 m Ethyl n-butyrate 8.5 mEthylene carbonate 14.7  m Ethylene dichloride 9.8 p Ethylene glycol14.6  s Ethylene glycol diacetate 10.0  m Ethylene glycol diethyl ether8.3 m Ethylene glycol dimethyl ether 8.6 m Ethylene glycol monobutylether 9.5 m (Butyl Cellosolve 

) Ethylene glycol monoethyl ether 10.5  m (Cellosolve 

) Furfuryl alcohol 12.5  s Glycerol 16.5  s Hexane 7.3 p Isopropylalcohol 8.8 m Methanol 14.5  s Methyl amyl ketone 8.5 m Methylenechloride 9.7 p Methyl ethyl ketone 9.3 m Methyl isobutyl ketone 8.4 mPropyl acetate 8.8 m 1,2-Propylenecarbonate 13.3  m Propylene glycol12.6  s Propylene glycol methyl ether 10.1  m Pyridine 10.7  s1,1,2,2-Tetrachloroethane 9.7 p Tetrachloroethylene 9.3 p(perchloroethylene) Tetrahydrofuran 9.1 m Toluene 8.9 p Water 23.4  s  

 

H-Bonding: p = poor; m = moderate; s = strong

indicates data missing or illegible when filed

TABLE 4 Table III: Solubility Parameters (δ) for Plasticizers andSolvents (Increasing δ value sequence) δ H-Bonding Solvent(cal/cm³)^(1/2)

Dimethlsiloxanes 4.9-5.9 p Diisodecyl phthalate 7.2 m Hexane 7.3 pDiamyl ether 7.3 m Diethyl ether 7.4 m Dioctyl phthalate 7.9 m Butylbutyrate 8.1 m Ethyl amyl ketone 8.2 m Ethylene glycol diethyl ether 8.3m Butal acetate 8.3 m Methyl isobutyl ketone 8.4 m Methyl amyl ketone8.5 m Amyl acetate 8.5 m Ethyl n-butyrate 8.5 m Ethylene glycol dimethylether 8.6 m Carbon tetrachloride 8.6 p Dioctyl sebacate 8.6 m Dioctyladipate 8.7 m Isopropyl alcohol 8.8 m Diethyl carbonate 8.8 m Propylacetate 8.8 m Diethyl ketone 8.8 m Dimethyl ether 8.8 m Toluene 8.9 pDi-n-hexyl phthalate 8.9 m Ethyl acetate 9.1 m Diamyl phthalate 9.1 mTetrahydrofuran 9.1 m Dibutyl sebacate 9.2 m Benzene 9.2 pTetrachloroethylene 9.3 p (perchloroethylene) Di(propylene glycol) 9.3 mmonomethyl ether Chloroform 9.3 p Dibutyl phthalate 9.3 m Methyl ethylketone 9.3 m Dibenzyl ether 9.4 m Ethylene glycol monobutyl ether 9.5 m(Butyl Cellosolve 

) Di(ethylene glycol) monobutyl 9.5 m ether (Butyl Carbitol 

) Chlorobenzene 9.5 p Methylene chloride 9.7 p Dipropyl phthalate 9.7 m1,1,2,2-Tetrachloroethane 9.7 p Ethylene dichloride 9.8 p Acetone 9.9 m1,2-Dichlorobenzene 10.0  p Diethyl phthalate 10.0  m Ethylene glycoldiacetate 10.0  m Di(propylene glycol) 10.0  s Carbon disulfide 10.0  p1,4-Dioxane 10.0  m Propylene glycol methyl ether 10.1  m Di(ethyleneglycol) monoethyl 10.2  m ether (Carbitol 

) Cresol 10.2  s Aniline 10.3  s Ethylene glycol monoethyl 10.5  m ether(Cellosolve 

) Pyridine 10.7  s Dimethyl phthalate 10.7  m N,N-Dimethylacetamide10.8  m Cyclohexanol 11.4  s Butyl alcohol 11.4  s Acetonitrile 11.9  pDimethyl sulfoxide 12.0  m Di(ethylene glycol) 12.1  sN,N-Dimethylformamide 12.1  m Furfuryl alcohol 12.5  s Propylene glycol12.6  s 1,2-Propylenecarbonate 13.3  m Methanol 14.5  s Ethylene glycol14.6  s Ethylene carbonate 14.7  m Glycerol 16.5  s Water 23.4  s  

H-Bonding: p = poor; m = moderate; s = strong  

 and Cellosolve are registered trademarks of Union Carbide Corp.

indicates data missing or illegible when filed

TABLE 5 Table IV: Solubility Parameters for Homopolymers 

Repeating Unit  

(Alphabetical Sequence) Acrylonitrile 12.5 Butyl acrylate  9.0 Butylmethacrylate  8.8 Cellulose 15.6 Cellulose acetate (56% Ac groups) 27.8Cellulose nitrate (11.8% N) 14.8 Chloroprene  9.4 Dimethylsiloxane  7.5Ethyl acrylate  9.5 Ethylene  8.0 Ethylene terephthalate 10.7 Ethylmethacrylate  9.0 Formaldehyde (Oxymethylene)  9.9 Hexamethyleneadipamide (Nylon  

) 13.6 n-Hexyl methacrylate  8.6 Isobornyl acrylate  8.21,4-cis-Isoprene  8.0 Isoprene, natural rubber  8.2 Isobutylene  7.8Isobornyl methacrylate  8.1 Isobutyl methacrylate  7.2 Laurylmethacrylate  8.2 Methacrylonitrile 10.7 Methyl acrylate 10.0 Methylmethacrylate  9.5 Octyl methacrylate  8.4 Propyl acrylate  9.0 Propylene 9.3 Propylene oxide  7.5 Propyl methacrylate  8.8 Stearyl methacrylate 7.8 Styrene  8.7 Tetrafluoroethylene  6.2 Tetrahydrofuran  9.4 Vinylacetate 10.0 Vinyl alcohol 12.6 Vinyl chloride  9.5 Vinylidene chloride12.2 (Increasing δ Value Sequence) Tetrafluoroethylene  6.2 Isobutylmethacrylate  7.2 Dimethylsiloxane  7.5 Propylene oxide  7.5 Isobutylene 7.8 Stearyl methacrylate  7.8 Ethylene  8.0 1,4-cis-Isoprene  8.0Isobornyl methacrylate  8.1 Isoprene, natural rubber  8.2 Laurylmethacrylate  8.2 Isobornyl acrylate  8.2 Octyl methacrylate  8.4n-Hexyl methacrylate  8.6 Styrene  8.7 Propyl methacrylate  8.8 Butylmethacrylate  8.8 Ethyl methacrylate  9.0 Butyl acrylate  9.0 Propylacrylate  9.0 Propylene  9.3 Chloroprene  9.4 Tetrahydrofuran  9.4Methyl methacrylate  9.5 Ethyl acrylate  9.5 Vinyl chloride  9.5Formaldehyde (Oxymethylene)  9.9 Methyl acrylate 10.0 Vinyl acetate 10.0Methacrylonitrile 10.7 Ethylene terephthalate 10.7 Vinylidene chloride12.2 Acrylonitrile 12.5 Vinyl alcohol 12.6 Hexamethylene adipamide(Nylon 6/6) 13.6 Cellulose nitrate (11.6% N) 14.8 Cellulose 15.6Cellulose acetate ( 

) 27.8  

indicates data missing or illegible when filed

In some embodiments, the host composition can be formed by crude oils,refined fuel, and in particular kerosene (e.g., Jet-A, Jet-A1, andmilitary fuel JP-8), gasoline, and diesel and other refined fuelsidentifiable by a skilled person.

As used herein the term “refined” can be considered to have its usualmeaning in the art. Thus, a refined hydrocarbon liquid composition isone that has been subjected to at least one process that is intended topurify it from a crude petroleum (crude oils/crudes) starting material.Thus, a refined fuel is a hydrocarbon liquid composition which hasundergone at least one process that can be considered to be adistillation, upgrading or conversion process, that is known to a personof skill in the art. Typically, a refined fuel is one that has undergonemore than one refining procedure in a refinery, such as a combination ofdistillation, upgrading and conversion. Therefore, in some instances therefined fuel composition can meet known, predetermined qualityparameters. In some instances, a refined hydrocarbon liquid compositioncan also include chemical additives that have been introduced to meetdesirable fuel specifications. Exemplary refined fuels comprise Jet Aand Jet A1 which are a kerosene-type aviation fuel comprising a mixtureof a large number of different hydrocarbons with carbon numberdistribution between about 8 and 16 (carbon atoms per molecule)identifiable by a skilled person. An additional exemplary refined fuelcomprise JP-8 or JP8 (for “Jet Propellant 8”) which is a kerosene typejet fuel, specified by MIL-DTL-83133 and British Defence Standard 91-87also identifiable by a skilled person. In particular, in someembodiments, the associative polymer can be selected depending on theregime of flows where drag reduction and/or flow rate enhancement isdesired as well as any other particular physical and/or chemicalproperties of the non-polar composition to be controlled.

In some embodiments the host composition can be formed by a mineral oil.The term “mineral oil” refers to various colorless, odorless, lightmixture of higher alkanes from a mineral source. In some embodiments,mineral oil can be a liquid by-product of refining crude oil to makegasoline and other petroleum products. This type of mineral oil is atransparent, colorless oil composed mainly of alkanes and cycloalkanes,related to petroleum jelly and has a density of around 0.8 g/cm³. Threebasic classes of mineral oils are alkanes, based on n-alkanes,naphthenic oils, based on cycloalkanes, and aromatic oils, based onaromatic hydrocarbons. Mineral oils can be in light or heavy grades, inwhich heavy grades mean higher viscosity. The viscosity of a mineral oilis correlated to its temperature, specifically, the higher thetemperature, the lower the viscosity.

In particular, in some embodiments, the chemical and/or physicalproperty can be controlled by controlling concentration of one or moreframing associative polymers in the host composition relative to theoverlap concentration c* of the one or more framing associative polymersin the host concentration. Accordingly one or more framing associativepolymers can be comprised in the host in a concentration of a fractionalor integer multiple of the overlap concentration (c*).

The terms “overlap concentration”, or “c*”, as used herein refer to theconcentration at which molecules of a non-associative form of theframing associative polymer (e.g. obtained from literature sources onthe backbone of interest or from experimental methods described hereinusing the polymer of interest modified to inactivate the functionalgroups to prevent association, for example by esterifying carboxylicacids or blocking carboxylic acid with triethylamine) dissolved in thehost begin to overlap each other, as opposed to being separated as theywould be in a more dilute solution. In particular, c* for particularpolymers in particular hosts can be identified by methods andcalculations identifiable to a skilled person (see, e.g. [21] andExample 23).

In particular, the chain length of the backbone can be chosen such thatthe backbone is long enough to ensure that a small concentration of thepolymer will suffice to produce a desired effect using relationshipsbetween chain length and the c* of the associative polymer describedherein. For example, a polymer that is effective at concentrations lessthan 1% by weight can be obtained by choosing a backbone length thatgives c* less than or approximately equal to 1% by weight. Inparticular, the relationship between chain length (e.g., expressed asthe weight-average molecular weight) and c* can be determined fromreferences identifiable by a skilled person or determined bycalculations as described herein.

In particular, for a non-associative polymer chain, the overlapconcentration is given by:

${c^{*} = \frac{3M_{w}}{4{\pi\left( R_{g}^{2} \right)}^{3/2}N_{a}}},$

wherein M_(w) is the weight-average molecular weight, R_(g) is theradius of gyration, and N_(a) is Avogadro's constant. The overlapconcentration represents a concentration equal to one polymer moleculeper spherical volume of radius R_(g), as illustrated for example in theexemplary schematic of FIG. 17. Throughout this disclosure, reference ismade to c* when describing the concentration of associative polymerrequired to achieve each type of desired chemical or physical property.Generally the pairings of polymer and host represent good solvent (e.g.a solvent in which the polymer-solvent interactions are morethermodynamically favorable than polymer-polymer interactions; see e.g.[22]) conditions for the polymer backbone. In good solvent conditions,R_(g) increases approximately as the ⅔ power of M_(w), so the expressionfor c* above shows that c* decreases as M_(w) increases. For a specificchoice of polymer backbone and host liquid, c* scales approximately as1/M_(w). For example, doubling the length of the polymer backboneapproximately reduces by half the concentration of associative polymerrequired to achieve a given effect.

In several exemplary embodiments, many polymers' data relating R_(g) toM_(w) are available for commonly used solvents [23]. When experimentalvalues are not available, an indicative estimate can be made using atheoretical chain model as herein described. For example, the estimateof R_(g) using the ideal chain model provides a conservative estimatec*of the concentration of polymer required to achieve a desired effect.A skilled person will realize upon a reading of the present disclosurethat the polymer backbone is in a good solvent condition when dissolvedin the host, so the actual c* of the polymer in the host can be lessthan the value of c* estimated using the ideal chain model.

For the purpose of selecting the degree of polymerization to use for thespan of the polymer (which is the backbone length in the simple case ofa linear telechelic structure), an equivalent expression can be writtenthat refers to tabulated parameters, including e.g. parameters availablefor many polymers. In particular, tabulated values of the characteristicratio, c_(∞), and the length and equivalent mass of a “Kuhn segment” (band M_(o)) can be used to estimate the chain length that will confer adesired effect with a selected concentration. For example, for mistcontrol, the polymer can be present at its overlap concentration. Inapplications in which a polymer concentration is desired to be at mostc_(max), a chain can be used that has sufficiently many Kuhn segments,N, so that the polymer begins to overlap when its concentration isapproximately c_(max) or less. Such chain can be given by:

$N^{3/2} = \frac{9\sqrt{6}}{2\pi\; b^{3}c_{\max}}$

where N is the number of Kuhn segments and corresponds to a linearpolymer (or span of a branched polymer) having molar mass NM_(o), whereM_(o) is the mass per Kuhn segment. Therefore, one can synthesize forexample a polymer that has a span of molar mass NM_(o)(and functionalgroups, selected with guidance below) and introduce the synthesizedpolymer to a composition at a concentration c* to provide mist control.A skilled person will realize that when using approximate expressionsfor c*, mist control is expected to improve by increasing or decreasingthe concentration relative to the estimated value of c*. In particular,in experiments that examine the extent of mist control with associativepolymer, concentrations of associative polymer of 0.5c* and 2c* can besuitable. Similar reasoning can be applied for other effects hereindescribed as will be understood by a skilled person.

A list of exemplary tabulated parameters is indicated below (Table 6;[24], p. 53):

TABLE 6 Characteristic ratios, Kuhn lengths, and molar masses of Kuhnmonomers for common polymers at 413 K Polymer Structure C 

 

 

 

1,4-Polyisopropene (PI) —( 

CH═CHCH( 

))— 4.6 8.2 0.830 113 1,4-Polybutadiene (PB) —( 

CH═CH 

)— 5.3 9.6 0.826 105 Polypropylene (PP) —( 

 

( 

))— 5.9 11    0.791 180 Poly(ethylene oxide) (PEO) —( 

 

O)— 6.7 11    1.064 137 Poly(dimethyl siloxane) (PDMS) —(OSi( 

) 

)— 6.8 13    0.895 381 Polyethylene (PE) —( 

 

)— 7.4 14    0.784 150 Poly(methyl methacrylate) (PMMA) —( 

C( 

)(COO 

))— 9.0 17    1.13  655 Atactic polystyrene (PS) —( 

CH 

 

)— 9.5 18    0.969 720

indicates data missing or illegible when filed

In addition, a skilled person can also identify the relationship betweenchain length and c* by experimental measurement, e.g. by measuring theshear viscosity of the host composition including the non-associativeform of the polymer as a function of the concentration of the polymer.

In particular, the overlap concentration of the backbone can bedetermined from conventional shear viscosity measurements of solutionscontaining various concentrations of the non-associative form of thepolymer. Alternatively, it can be evaluated using the weight averagemolecular weight of the longest span of the polymer, which is oftencharacterized as part of the synthesis and purification of a syntheticpolymer.

In particular, c* can be determined at a given temperature by measuringthe viscosities of a non-associative polymer in an appropriate host atvarying concentrations using a rheometer wherein at c* a deviation fromlinearity is observed in the plot of viscosity versus polymerconcentration. Linear regression is performed on the data from bothdilute and concentrated regimes, and the crossover of the two linearfits represents the overlap concentration, c* (see, e.g. [24, 25] andFIG. 38).

In particular, in some embodiments, a way to identify a “desired overlapconcentration” is to consider the type of beneficial effect that isneeded. For example, for a desired effect of mist control, aconcentration of polymer can be used that is approximately equal to theoverlap concentration. In particular, in embodiments herein describedwhere control of drag reduction and/or flow rate enhancement and relatedduration is desired, a concentration range of the associative polymercan be selected between from about 0.001 c* to 1c*, depending on theextent drag reduction desired alone or in combination with anotherphysical and/or chemical property to be controlled.

In embodiments where control of additional physical or chemical propertyis desired the specific c* value can be selected taking into account thec* values associated with the control of the additional physical and/orchemical property.

For example a concentration range suitable for mist control can bebetween 0.5c* to 2c*. In embodiments in which a desired effect isenhancing fuel efficiency, a polymer concentration can be used in thenon-polar compositions herein described that is less than c*, and inparticular can be between 0.1c* and 0.5c*. In embodiments in which thedesired effects are drag reduction and enhanced lubrication, a polymerconcentration can be a concentration below or approximately equal c*,and in particular can be between 0.05c* to c*. In embodiments in which adesired effect is converting a liquid into a gel, a concentrationgreater than c* can be provided and in particular a concentration from2c* to 10c*.

Selection of one or more specific associative polymers that can becomprised within the composition at a concentration relative to the c*selected to control a set of one or more chemical and/or physicalproperties can be performed in view of the characteristics of functionalgroups, chain structures, and weight average molecular weight ofassociative polymers herein described.

In some embodiments, the functional groups described herein at the endsof the backbone of the associative polymer can be selected to ensureassociation occurs with the range of the polymer concentrationsselected. In conjunction with the selection of functional groups, thesynthetic chemistry is selected to be appropriate for introduction ofsuch groups.

A skilled person will realize that characteristics of the host thatinfluence the selection of functional groups include, for example, itsdielectric constant and whether or not it contains protic species orspecies that offer a lone pair of electrons. Non-polar liquids generallycontain molecules made mainly of atoms with similar electronegativities,such as carbon and hydrogen (for example, hydrocarbons that dominatefuels and many lubricants). Bonds between atoms with similarelectronegativities lack partial charges, making the moleculesnon-polar. A common way of quantifying this polarity is the dielectricconstant. A skilled person will also realize that another characteristicof components in the host liquid is whether or not they have O—H or N—Hbonds that can participate in hydrogen bonding. A skilled person wouldrecognize these as protic molecules. Examples of protic species that maybe present in host liquids in the disclosed ranges of dielectricconstants include, for example secondary amines with substantialhydrocarbon content (e.g., Diisobutylamine, which has dielectricconstant 2.7; dipropylamine, which has dielectric constant 2.9;Methylbenzylamine, which has dielectric constant 4.4), carboxylic acidswith substantial hydrocarbon content (e.g., palmitic acid, which hasdielectric constant 2.3; linoleic acid, which has dielectric constant2.6; oleic acid, which has dielectric constant 2.5), and alcohols withsubstantial hydrocarbon content (e.g., hexadecanol, which has dielectricconstant 3.8). In addition, a skilled person will also realize thatother protic species (e.g., protic species that in their pure state canhave a dielectric constant greater than 5, such as aniline and phenol)can be present as minor species in a host liquid that has dielectricconstant less than 5.

A skilled person will realize that another relevant characteristic ofcomponents in the host liquid is whether or not they present a lone pairof electrons that can participate in hydrogen bonding. Examples ofspecies with lone pairs that may be present in host liquids in thedisclosed ranges of dielectric constants include alkyl-quinoxalines(e.g., 2,3-Dimethylquinoxaline, which has dielectric constant 2.3),tertiary amines (e.g., triethylamine, which has dielectric constant 2.4)and nonconjugated esters (e.g., isoamylvalerate, which has dielectricconstant 3.6). In addition, a skilled person will also realize thatother lone-pair species (that in their pure state might have adielectric constant greater than 5, such as pyridine andmethylethylketone) can be present as minor species in a host liquid thathas dielectric constant less than 5. In addition, a skilled person willrealize that components that are used as additives when the host liquidis formulated can also be present. For example, metal chelating agents(e.g., N,N-Disalicylidene-1,2-propanediamine) can be present in a hostliquid that is a fuel. A skilled person will realize that the presenceof these constituents influences the selection of functional groupsdepending on the presence of protic species or species that offer a lonepair of electrons as described herein.

A skilled person will also realize the presence of protic species can,in some circumstances, interfere with FG, and in particular with FGa,association mediated by hydrogen bonding. The skilled person willrealize that one way to overcome the interference is to increase thenumber of hydrogen bond moieties at the chain ends. The skilled personwill also realize that another way to overcome the interference is toreduce the concentration of protic species in the host. A skilled personwould recognize that these two approaches can be used together. Inaddition, a skilled person will also realize that, all other factorsbeing equal, increasing the dielectric constant of the host weakens theinteraction (e.g., conventional hydrogen bonds, charge-assisted hydrogenbonds, charge transfer interaction, metal-ligand interactions). Forexample, increasing the dielectric constant from 2.4 (toluene) to 4.8(chloroform) decreases the association constant for the Hamiltonreceptor and cyanuric acid by an order of magnitude. Accordingly, FGasthat provide a stronger association (e.g., charge-assisted hydrogenbonding or a metal-ligand interaction) are expected to be beneficialwhen the dielectric constant is greater than 2.5. A skilled person wouldrealize that the selection of FGas that provide strong association canbe used together with increasing the number of associative groups at thechain ends and with reducing the concentration of host components thathave high dielectric constants.

In particular, in some embodiments, the value of the concentration ofthe associative polymer relative to overlap concentration c* can begoverned by the selection of chain-host pair and can be insensitive tothe specific choice of FGa. A skilled person will understand that theoverlap concentration can vary with temperature, in a manner that isparticular to a specific chain-host pair. For example, the selection ofpolymer backbone and host governs the solvent quality; and, for a givensolvent quality, the degree of polymerization is chosen to adjust c*once the chain-host pair is selected. In this connection selecting agreater degree of polymerization, provides a greater R_(g) and,consequently, a reduced c* as will be understood by a skilled person.

In some embodiments herein described, the chain structure between thenodes (e.g. the chain being a polyolefin, polydiene, or other structureidentifiable to a skilled person upon a reading of the presentdisclosure) can be chosen such that it interacts favorably with thehost, the state of the backbone can be estimated using good solvent(e.g. a solvent in which the polymer-solvent interactions are morethermodynamically favorable than polymer-polymer interactions; see e.g.[22]) scaling for its pervaded volume. Over most of the molecular weightrange of interest, the ideal chain approximation (e.g. approximation ofthe polymer chain as a random walk and neglecting any kind ofinteractions among monomers; see e.g. [24]) can also be useful: itprovides a lower bound on R_(g) that is usually within a factor of 2 ofthe good solvent chain dimensions, as shown in FIG. 16 for the case ofpolystyrene for a good solvent such as toluene, and a theta solvent(e.g. a solvent in which the polymer-solvent interactions areapproximately as equally thermodynamically favorable as polymer-polymerinteractions; see e.g. [22]) such as cyclohexane. In particular, thevalue of the radius of gyration can be used to estimate theconcentration at which polymer molecules would begin to overlap oneanother: the overlap concentration c* corresponds to the value thatgives approximately one polymer molecular per (R_(g) ²)^(3/2).

Additional factors related to applications of the resulting compositions(e.g. distribution through a pipeline, storage for a certain time periodand other factors identifiable by a skilled person), can also be takeninto account in the selection of the specific associative polymer orcombination thereof and/or in the selection of the related concentrationin the host composition relative to c* within a range associated tocontrol of one or more chemical and/or physical properties.

In embodiments in which a low concentration of polymer is desired, areduction in the concentration of the associative polymer relative to c*can be obtained by selecting a polymer with high degree ofpolymerization. In some of those embodiments, the degree ofpolymerization of the polymer is low enough that the polymers do notdegrade during necessary handling. For example, in embodiments in whichthe non-polar compositions are fuel or other liquid and the liquid isintended to travel through a distribution system, minimization of thedegradation of the polymer upon passage through pumps and filters,and/or minimization of degradation during turbulent flow in transportpipelines or hoses can be desirable. In this connection, in exemplaryembodiments in which the polymers comprise linear chains, keeping theweight-average molar mass below 1,000,000 g/mol can give adequatestability with respect to shear degradation. In exemplary embodiments inwhich the polymer comprises lightly branched molecules, havingnode-chain-node segments that are individually greater than 10,000g/mol, the longest span of the molecule can be kept below the thresholdfor shear degradation (typically less than 1,000,000 g/mol).

In embodiments wherein conversion of liquid to gel is desired, asolution or gel that has dielectric constant less than 5 and comprises apolymer that has weight average molar mass between 100,000 g/mol and1,000,000 g/mol, can comprise the polymer at a concentration that isbetween 0.1c* and 10 c*. The specific concentration can be determinedbased on the measured length and backbone composition of the polymer,and the polymer molecules manifestly associate with one another asevidenced by shear viscosity that is anomalously enhanced relative to anon-associative polymer of the same molar mass and backbone structure orby light scattering showing structures that are much larger than anon-associative polymer of the same molar mass and backbone structure.The latter measurements can be performed for example by removing thepolymer from the composition and reconstituting them in a solvent thathas a dielectric constant that is close to that of the composition(±20%) at a concentration of c* based on the weight-average molecularweight determined by GPC equipped with multi-angle static lightscattering.

In some embodiments when the concentration of the framing associativepolymer is equal to or lower than 0.02 c* the associative polymer canhave a weight-average molecular weight equal to or higher than10,000,000 g/mol. In some of those embodiments, the associative polymercan be used for drag reduction in the non-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between than 0.05 c* to 0.1 c*the associative polymer canhave a weight-average molecular weight equal to or higher than10,000,000 g/mol In some of those embodiments, the associative polymercan be used for mist control in the non-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between 0.02 c* and 0.05 c*, the associative polymer can havea weight-average molecular weight between 2,000,000 g/mol to 10,000,000g/mol In some of those embodiments, the associative polymer can be usedfor drag reduction in the non-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between 0.05 c* and 0.1c*, the associative polymer can have aweight-average molecular weight between 500,000 g/mol to 2,000,000g/mol, and in particular 1,000,000 g/mol to 2,000,000 g/mol. In some ofthose embodiments, the associative polymer can be used for dragreduction and/or mist control in the non-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between 0.1 c* and c*, the associative polymer can have aweight-average molecular weight between 400,000 g/mol to 1,000,000 g/molIn some of those embodiments, the associative polymer can be used fordrag reduction and/or mist control in the non-polar composition. Inparticular, when the weight-average molecular weight is at least 400,000g/mol the associative polymer can be used at concentration is between0.1c* and 0.5c* for drag reduction of the host composition; at aconcentration of about 0.5c* for drag reduction and possibly for mistcontrol of the host composition depending on the molecular weight of thepolymer, and at a concentration of less than approximately c* for dragreduction and mist control of the host composition.

In some embodiments when the concentration of the framing associativepolymer is between 0.5 c* and c*, the associative polymer can have aweight-average molecular weight between 400,000 g/mol to 1,000,000g/mol. In some of those embodiments, the associative polymer can be usedfor drag reduction, mist control and/or lubrication in the non-polarcomposition.

In some embodiments when the concentration of the framing associativepolymer is between c* and 2c*, the associative polymer can have aweight-average molecular weight between 400,000 g/mol to 1,000,000g/mol. In some of those embodiments, the associative polymer can be usedfor mist control, lubrication, and/or viscoelastic properties of thenon-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between c* and 2c*, he associative polymer can have aweight-average molecular weight between 100,000 g/mol to 400,000 g/mol.In some of those embodiments, the associative polymer can be used fordrag reduction, lubrication and/or viscoelastic properties of thenon-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between c* and 3c*, the associative polymer can have aweight-average molecular weight between 400,000 g/mol to 1,000,000g/mol. In some of those embodiments, the associative polymer can be usedfor mist control, lubrication and/or control of viscoelastic propertiesin the non-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between c* and 3c*, the associative polymer can have aweight-average molecular weight between 100,000 g/mol to 400,000 g/molIn some of those embodiments, the associative polymer can be used forlubrication, and/or control of viscoelastic properties in the non-polarcomposition.

In some embodiments when the concentration of the framing associativepolymer is between 2c* to 10c*, the associative polymer can have aweight-average molecular weight between 100,000 g/mol to 400,000 g/molIn some of those embodiments, the associative polymer can be used forlubrication, and/or viscoelastic properties of the non-polarcomposition.

In some embodiments when the concentration of the framing associativepolymer is between 2c* to 10c*, the associative polymer can have aweight-average molecular weight between 100,000 g/mol to 1,000,000 g/molIn some of those embodiments, the associative polymer can be used forlubrication, and/or viscoelastic properties and in particulargelification of the non-polar composition.

In some embodiments when the concentration of the framing associativepolymer is between 3c* and 10c*, the associative polymer can have aweight-average molecular weight between 100,000 g/mol to 1,000,000 g/molIn some of those embodiments, the associative polymer can be used forlubrication, and/or control of viscoelastic properties and in particulargelification in the non-polar composition.

In embodiments in which the composition comprise liquid fuels, such asgasolines, diesel fuels, kerosene and jet fuels, such compositions cancomprise polymers with molar mass between 100,000 g/mol and 1,000,000g/mol having backbones that, as bulk polymers, have dielectric constantless than 3 and are present in the composition at a concentration thatis between 0.1c* and 10c*, based on the measured weight-average molarmass and backbone composition of the polymer, and the polymer moleculesmanifestly associate with one another as evidenced by shear viscositythat is enhanced relative to a non-associative polymer of the same molarmass and backbone structure or by light scattering showing structuresthat are much larger than a non-associative polymer of the same molarmass and backbone structure. The latter measurements can be performedfor example by removing the polymer from the composition andreconstituting them in toluene at a concentration of c* based on theweight-average molar mass determined by GPC equipped with lightscattering. In several examples of the current disclosure toluene isindicated as a reference host because it has a dielectric constant ofapproximately 2.2, which is at the upper range of diverse fuels and,therefore, gives a conservative diagnostic of association. That is, apolymer that forms intermolecular associations in toluene will formintermolecular associations in gasoline, diesel, kerosene and jet fuel,among others.

In some embodiments, polymer for improving fuel efficiency can beeffective at 10000 ppm or less with weight average molecular weightbelow 1,000,000 g/mol, possibly after more than 10 passages of the fuelthrough a fuel pump. In some embodiments, associative polymers canremain uniformly dissolved for at least 2 weeks or even months even at−30° C.

In some embodiments, with weight average molecular weight 400,000 g/molchains, droplet behavior of non-polar composition comprising associativepolymers herein described is expected to match 4,200,000 g/mol (weightaverage) polyisobutylene, a commonly used standard material to achievemist control effect using high molecular weight polymer, compared at thesame, concentration of 0.3%.

In some embodiments, if for a particular application the polymerconcentration is desired to be kept low, this can be achieved byincreasing the length of the polymer chain between associative groups.The reason for this is that polymers tend to adopt compact conformationsin isolated clusters when the concentration is far below their overlapconcentration; increasing the length of the polymer between associativegroups decreases the overlap concentration, thereby allowing desiredproperties to be achieved with a lower concentration of polymer.

In some embodiments, if for a particular application the polymeradditive is desired to survive passage through pumps and turbulent pipeflow, this can be achieved by keeping the length of the polymer belowthe threshold at which chain scission occurs in intense flows. For anumber of polymers, the literature provides values of the chain lengthabove which chains scission occurs (e.g. polyisobutylene) For any choiceof polymer backbone structure, the threshold length (or equivalently,degree of polymerization or molar mass) above which chain scissionoccurs upon passage through pumps or turbulent pipe flow can bedetermined as will be understood by a skilled person.

In some embodiments, for the purpose of creating additives that delivervaluable effects at low polymer concentration, use of chain segmentshaving molar mass between 100,000 g/mol and 500,000 g/mol between FG,and in particular FGa, and node can be desired. This range of structurescan associate at low concentrations to give desired properties. Forexample, in the context of fuels, the resulting polymers can inhibitmisting in order to reduce the risk of post-crash fires; can controlatomization to increase fuel efficiency and/or reduce emissions; canconfer drag reduction that reduces pumping costs and improves throughputthrough existing pipelines; and improve lubrication. In particular,polymers of the present disclosure can survive prolonged, severe shearwith little degradation; the polymers do not interfere with filteringfuel; the polymers do not interfere with dewatering fuel.

According to the above indication and to the additional indicationprovided in the disclosure, in some embodiments, one skilled in the artcan identify whether or not a host of interest (e.g., a particularlubricant oil) is suitable for application of the associative polymersbased on the dielectric constant of the host, and the skilled person canidentify suitable monomer structures using knowledge of the dielectricconstant or solubility parameter of the resulting polymer, and thusselect the degree of polymerization (e.g. by synthesizing a polymerbackbone of a particular weight-averaged molecular weight) to achieve adesired c*.

In some embodiments herein described once the suitability of a potentialhost is determined, as well as the selection of the monomer and theselection of the degree of polymerization are made, functional groupscan be selected that are able to associate according to the indicatedassociation constant. In particular, in some embodiments when the hosthas a relatively low dielectric constant (e.g. ε<2) and little or noparticipation in hydrogen bonds, there are many associative groups thatare effective as will be understood by a skilled person. Therefore,secondary considerations can be applied to narrow down the selection(such as cost, sensitivity to ionic species, nature of combustionproducts, and other considerations identifiable to a skilled person).For example, in some instances, with increasing dielectric constant ofthe host, many of the useful interactions (hydrogen bonding, chargetransfer, acid-base, and others identifiable to a skilled person) becomeprogressively weaker. Therefore, clusters of functional groups may berequired to confer adequate association. Consequently, for solvents thathave dielectric constant greater than 2.5, dendrimeric FG can be usedthat include multiple associative groups (examples are shown for FG thateach present four or eight copies of a chosen associative group).

For example, in embodiments herein described where drag reduction (e.g.the flow resistance of a non-polar composition through a conduit such asan oil pipeline or fuel line in a vehicle) is the property sought to becontrolled, a skilled person can identify the solubility parameter ofthe fluid, and then can identify polymer backbones that aresubstantially soluble in the fluid (e.g. by comparing the solubilityparameters and/or using the solubility parameters to determine theFlory-Huggins interaction parameter as described herein). The selectionof particular polymers for the backbone of the associative polymersuitable to be included at a concentration relative to c* below c* canbe further refined based on, for example, on the cost of the polymers,or the ease and/or expense of the polymerization chemistry, as would beidentifiable to a skilled person.

In particular, for drag reduction, a skilled person would realize it canbe desirable to minimize the amount of polymer used for two reasons: tominimize cost and to avoid undue increase in the shear viscosity of themixture. Accordingly, the length (expressed as the weight-averagedmolecular weight) of the backbone of the associative polymer can be nearthe threshold imposed by shear degradation, which a skilled person wouldunderstand to be in the range of approximately 500,000 g/mol forhydrocarbon polymers such as polyisobutylene, polybutadiene,polyolefins, and others identifiable to a skilled person.

In particular, a skilled person can verify that the chain lengthselected resists shear degradation by performing analyses known to theskilled person. For example, the viscosity of a non-polar compositioncomprising the associative polymers described herein can be measuredbefore and after recirculation through a conduit (e.g. by using a fuelpump to recirculate a sample of the non-polar composition) anddetermining if there is a difference in viscosity between the two timepoints (e.g., if the viscosity decreases after recirculation, theassociative polymer can be considered to have undergone sheardegradation).

As another example, if mist control is among the properties of thenonpolar composition desired to be controlled, the polymer backboneselection among the possible polymers to be included at a concentrationrelative to c* between 0.5c* to 2c* can be based on solubility of the innonpolar composition as described herein (e.g. solubility parametersand/or Flory-Huggins interaction parameter), with the additionalconsideration of the associative polymer having negligible effect on thecalorific value of the nonpolar composition in which mist control isdesired, as would be identifiable to a skilled person (e.g. by using thecalorimetric method ASTM D240-09). The functional groups describedherein at the ends of the backbone of the associative polymer can bechosen to ensure that association occurs at desired concentration suchthat heteroatom content is so low as to not affect burning. For example,association can be measured using titration techniques identifiable to askilled person (see, e.g., [26]). Using the titration methods, theskilled person can identify a concentration at which the particularassociative polymers (with a given number of end groups containingheteroatoms) associate; if the concentration is suitable based on c*considerations (e.g. the particular concentration of the associativepolymer relative to c* to control a particular property such as mistcontrol) the skilled person can then measure the calorific value usingASTM D240-09. If the concentration is not suitable, the number of endgroups can be changed accordingly (e.g. by increasing the number forgreater association at a given concentration, or by decreasing thenumber for lesser association), the titration re-performed, and thecalorific value re-measured.

In various embodiments, associative polymers herein described can bemade with methods in which a backbone polymer is provided which is thenfunctionalized with suitable FGs and in particular with FGas.

In some embodiments, in which the backbone has a structural unit offormula -

node

-

chain]

- (II), wherein

-   -   chain is a non-polar polymer substantially soluble in a        non-polar composition, the polymer having formula

R₁-[A]_(n)R₂  (III)

-   -   in which        -   A is an organic moiety forming the monomer of the polymer;        -   R₁ and R₂ are independently selected from any carbon based            or organic group; and        -   n is an integer ≥1; and    -   node is a chemical moiety covalently linking one of R₁ and R₂ of        at least one first chain with one of the R₁ and R₂ of at least        one second chain;    -   and wherein the chain and node of different structural units of        the polymer can be the same or different and the polymer        presents two or more terminal R₁ and R₂ groups        the method can comprise: providing the polymer having structural        unit of formula -        node        -        chain]        - (II) and attaching functional groups FG herein described to        terminal R₁ and/or R₂ groups of the polymer.

In some embodiments, an associative polymer can be provided by forming apolymer chain through a method of polymerization of a suitable monomersuch as those described in [18], so that the desired architecture(linear, branched, star, and other architectures identifiable to askilled person) is generated and individual polymer chains aresubstantially terminated by chemical groups that are amenable tofunctionalization. The end groups can already be functionalized by FGsand in particular FGas or formed by precursors that are converted toFGs, and in particular FGas (e.g., by deprotection or functional groupsthat are suitable for covalent attachment of FGs). This prepolymer canthen be reacted with a molecule containing the desired FG, so that FGsare introduced to the polymer chain through chemical transformationscommonly described as functional group interconversions. Thus, in someembodiments the desired polymer composition can be achieved in atwo-step process, in which after the first step reaction of the monomergives a polymer that does not substantially include the desired FG orFGs, which are introduced in the second step. For example, theprepolymer may be synthesized as substantially terminated withfunctional groups known in the art to be “leaving groups” such ashalide, triflate or tosylate, and the desired FG or FGs introduced tothe polymer chain through nucleophilic substitution reaction.

In some embodiments, suitable monomers comprise dienes, olefins,styrene, acrylonitrile, methyl methacrylate, vinyl acetate,dichlorodimethylsilane, tetrafluoroethylene, acids, esters, amides,amines, glycidyl ethers, isocyanates, and mixtures of these.

In some embodiments, the associative polymer suitable for drag reductioncan be selected based on the Reynolds number of the host composition inthe flow pattern where the control is desired, wherein when the Reynoldsnumber of the host composition is in the range of about 5,000≤Re≤25,000or possibly up to 1,000,000 Re, the association constant (k) is in therange of 4≤log₁₀ k≤12; and when the Reynolds number is in the range ofabout Re 25,000, the association constant (k) is in the range of 6≤log₁₀k≤14.

In some embodiments, associative polymers that can be used for dragreduction in flow having Reynolds numbers equal to or higher than 5000comprise one or more of a telechelic 1,4-PB polymer with each end-grouphaving one tertiary amine group (Di-MB), a telechelic 1,4-PB polymerwith each end-group having two tertiary amine groups (Di-DB) atelechelic 1,4-PB polymer with each end-group having four tertiary aminegroups (Di-TB), a telechelic 1,4-PB polymer with each end-group havingeight tertiary amine groups (Di-OB), a telechelic 1,4-PB polymer witheach end-group having one carboxyl groups (Di-MA), a telechelic 1,4-PBpolymer with each end-group having two carboxyl groups (Di-DA), atelechelic 1,4-PB polymer with each end-group having four carboxylgroups (Di-TA), a telechelic 1,4-PB polymer with each end-group havingeight carboxyl groups (Di-OA), a telechelic 1,4-PB polymer with eachend-group having one tert-butyl ester groups (Di-ME), a telechelic1,4-PB polymer with each end-group having two tert-butyl ester groups(Di-DE), a telechelic 1,4-PB polymer with each end-group having fourtert-butyl ester groups (Di-TE), and/or a telechelic 1,4-PB polymer witheach end-group having eight tert-butyl ester groups (Di-OE). Inparticular in those embodiments the molecular weight of the polybutenecan have any values among the ones described, e.g. an overall weightaverage molecular weight, M_(w), equal to or lower than about 2,000,000g/mol, and/or a Mw equal to or higher than about 100,000 g/mol.

In some embodiments, associative polymers that can be used for dragreduction in flow having Reynolds numbers equal to or higher than 5000comprise the following pairs: Di-TA/Di-MB (1 tertiary amine),Di-TA/Di-DB, Di-TA/Di-TB; Di-TB (4 tertiary amines)/Di-MA, Di-TB/Di-DA;Di-OB(8 tertiary amines)/Di-MA, Di-OB/Di-DA, and Di-OB/Di-TA.

The association polymers described herein can be synthesized by methodsknown to a skilled person. In particular, following selection of abackbone with a desired contour length L and Mw the backbone can bemanufactured with methods known to a skilled person. For example, thebackbone can be synthesized by Ring-Opening Metathesis Polymerization(ROMP) chemistry and functionalized at the ends of the backbone usingappropriate chain transfer agents (see, e.g., Examples section hereinand [27]). In addition, anionic polymerization, Atom-transferRadical-Polymerization (ATRP), Reversible Addition-Fragmentation chainTransfer polymerization (RAFT) and other polymerization techniquesidentifiable to a skilled person (including an alternative overview ofmetathesis techniques) can be used to synthesize several types ofbackbones (e.g. block, star, branched and other architectures) andintroduce of many different types of functional groups at the ends ofthe polymer chain (or elsewhere if desired) (see, e.g. [28, 29]).

In certain embodiments, an associative polymer in accordance with thepresent disclosure can be provided by forming a polymer chain such thatthe desired architecture is generated, and individual polymer chains aresubstantially terminated by the desired FG, in situ. Thus, in someembodiments the desired polymer composition can be achieved in a singlestep process, and reaction of the monomer affords a polymer thatincludes the desired FG or FGs. In yet other embodiments, the desiredFGs can be introduced to the polymer chain in a form such that theultimate function of such FGs is masked by a chemical substitution (e.g.the FGs feature one or more “protecting groups”), and the desiredfunctionality of the FGs can then be enabled for example through removalof such a “protecting group” through chemical transformation insubsequent steps. However, in some embodiments, the desired polymercomposition can still be achieved in a single step process, and thepolymer as synthesized includes the desired FG or FGs in protected form.In some of those embodiments, suitable monomers include cyclic olefinsand acyclic α,ω-dienes.

Suitable methods of polymerization in accordance with some embodimentsherein described, comprise ring-opening metathesis polymerization (ROMP)and acyclic diene metathesis polymerization (ADMET), in the presence ofsuitable chain transfer agent (CTA) typically consisting of the FGsuitably disposed about a reactive olefinic functionality (e.g.cis-double bond). The FG or FGs can be in their ultimate functional formin this CTA, or can be in “protected” form such that unmasking of theultimate functional form may be achieved through removal of this“protecting group” through chemical transformation.

Suitable “protecting groups” in accordance with some embodiments hereindescribed, comprise those described in “[30].

For example, in some embodiments where the polymer backbone is made by aROMP polymerization (e.g. using cyclooctadiene to synthesize a backboneof repeating ═CHCH₂CH₂CH═CHCH₂CH₂CH=units), the ends of the polymerbackbone can be functionalized with appropriate chain transfer agents toprovide functionalized ends of the backbone which can be furthertransformed to provide functional groups capable of being correspondingfunctional groups, as shown for example in Examples 1-3 where carboxylicacid functional groups are installed. A skilled person will realize upona reading of the present disclosure that analogous reactions can beperformed to synthesize other backbones such as poly(vinylacetate) (e.g.RAFT polymerization as shown, for example in [31]; or free radicalpolymerization of vinyl acetate using a free radical initiatorcomprising FG groups as shown, for example, in [32]).

In particular, as exemplified in Example 3, chain transfer agents can beused to attach moieties substituted with chloro groups, which can thenbe displaced with azide groups (e.g. using trimethylsilyl (TMS) azide bymethods identifiable to a skilled person). A moiety comprising attachedalkyne groups can then be reacted with the azide groups via reactionssuch as the azide-alkyne Huisgen cycloaddition (e.g. click reaction) toattach the moiety to thereby attach the FG to the backbone (see, e.g.Example 3).

In yet further embodiments, an associative polymer in accordance withthe present disclosure can be provided by metathesis applied to a highmolecular weight (M_(w)>5,000,000 g/mol) poly(diene) such aspoly(butadiene) in the presence of suitable CTA and metathesis catalystto give a shorter poly(diene) substantially terminated by an FG and inparticular FGas, with the diene:CTA ratio chosen to afford the desiredmolecular weight for the product telechelic polymer.

In particular methods of these particular embodiments, the starting highmolecular weight poly(diene) can be linear and substantially free of1,2-vinyl groups in the polymer backbone.

In exemplary methods to make a polymer of the present disclosure, thepolymer can be made by ROMP in a continuous process. In particular,methods of these particular embodiments the continuous process can usereactions in series (FIG. 10). In relation to compositions that are usedas liquid fuels the continuous production of the associative polymersherein described can be performed near or inside a petrochemicalrefinery and incorporated into a product continuously.

In exemplary methods to make a polymer of the present disclosure, thepolymer can be made by ring-opening metathesis polymerization (ROMP) toobtain desired end-functional telechelic polymers of weight-averagemolecular weight 100,000 to 1,000,000 g/mol.

In exemplary methods to make a polymer of the present disclosure, thepolymer can be made by related polymerization and/or functionalizationmethods to make functional telechelics of molecular weight 100,000 to1,000,000 g/mol.

In some embodiments, a mixture of framing associative polymers andcapping associative polymers are produced simultaneously.

In various embodiments, associative polymers herein described can beused in methods and systems to control physical and/or chemicalproperties of an associative non-polar composition in a flowcharacterized by a Reynolds number Re, and a characteristic length d, inparticular to obtain a controlled drag reduction and/or flow rateenhancement effect alone or in combination with other physical and/orchemical properties of the associative non-polar composition in the flowas herein described.

The method comprises providing a host composition having having aviscosity μ_(h), a density ρ_(h) and a a dielectric constant equal to orless than about 5 and providing at least one framing associative polymersubstantially soluble in the host composition; and combining the hostcomposition and the at least one framing associative polymer hereindescribed at a concentration c between from about 0.01c* to about 10c*selected based on the molecular weight of the at least one framingassociative polymer (and/or radius of gyration) and on a physical and/orchemical property and in particular rheological property to becontrolled.

In the method, the longest span of the at least one framing associativepolymer has a countour length ½L_(bf)≤L_(f)<L_(bf), wherein L_(bf) is arupture length of the at least one framing associative polymer innanometers when the at least one framing associative polymer is withinthe host non-polar composition at a concentration c to provide theassociative non-polar composition in a flow, L_(b) being given byimplicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho\; d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.

In embodiments when the selected when c≤2c*, μ is μ_(h), and ρ is ρ_(h),and when c>2c*, μ is the viscosity of the associative non-polarcomposition μ_(a), and ρ is the density of the associative non-polarcomposition μ_(a).

In embodiments where the capping associative polymer is provided, themethod further comprises combining the at least one capping associativepolymer in the non-polar composition in an amount up to 20% of a totalassociative polymer concentration of the non-polar composition.

In some embodiments, the method can further comprise selecting aconcentration c of the at least one framing associative polymer in thehost composition, the concentration depending on the averaged molecularweight and/or radius of gyration of the at least one framing associativepolymer and on a physical and/or chemical property to be controlledbased on the factors herein described before the combining. A skilledperson will be able to select the specific Mw, Radius of gyration andconcentration of the at least one framing associative polymer in thehost composition in view of the present disclosure.

In the method combining the at least one framing associative polymer andoptionally the at least one capping associative polymer is performed toobtain the associative non-polar composition. The method also comprisesapplying forces to the associative non-polar composition to obtain aflow characterized by the Reynolds number Re, and the characteristiclength d.

In embodiments, herein described applying forces can be performed byapplying mechanical forces to transfer mechanical energy into theassociative non-polar composition to become kinetic energy of thecomposition and resulting in a flow of the associative non-polarcomposition. For example in a pipeline a Vane pump can provide frictionforces in the enclosed space of the pipeline, which is transferred to anassociative non-polymer composition in the pipeline to create the flow.In embodiments, herein described presence and concentration of framingassociative polymers will allow to control one or more rheologicalproperties of the associative non-polar composition in the flow.

In particular in exemplary embodiments, framing associative polymer canbe used, alone or in combination with capping associative polymers, in amethod to control resistance to flow and/or flow rate enhancement of anon-polar composition in a flow characterized by a Reynolds number Re,and a characteristic length d. In some of those embodiments additionalphysical and/or chemical property of the non-polar composition can alsobe controlled. The method comprises providing a host composition havinga viscosity μ_(h), a density ρ_(h) and a dielectric constant equal to orless than about 5; and providing at least one framing associativepolymer substantially soluble in the host composition and having aweight-average molecular weight equal to or higher to 200,000 g/mol. Themethod comprises combining the host composition and the at least oneframing associative polymer herein described at a concentration cbetween from about 0.01c* to about 1c* selected based on the molecularweight of the at least one framing associative polymer and on a physicaland/or chemical property and in particular rheological property to becontrolled.

In particular, in the method, the longest span of the at least oneframing associative polymer has a countour length ½L_(bf)≤L_(f)<L_(bf),wherein L_(bf) is a rupture length of the at least one framingassociative polymer in nanometers when the at least one framingassociative polymer is within the host non-polar composition at aconcentration c to provide the associative non-polar composition in aflow, L_(bf) being given by implicit function

$F_{bf} = {\frac{{\pi\mu}_{h}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho_{h}d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which in which F_(bf) is the rupture force of the framing associativepolymer in nanonewtons, Re is the Reynolds number of the flow, d is thecharacteristic length of the flow in meters, μ_(h) is the viscosity ofthe host non-polar composition in Pa·s, and ρ_(h) is the density of thehost non-polar composition in kg/m³.

In some embodiments, the method can further comprise selecting aconcentration c of the at least one associative polymer in the hostcomposition between from about 0.01c* to about 1c* depending on theaveraged molecular weight of the at least one associative polymer and ona physical and/or chemical property to be controlled based on thefactors herein described before the combining.

In some embodiments, the method can further comprise determining anoverlap concentration c* for the at least one framing associativepolymer before performing the selecting;

In some embodiments, the non-polar composition resulting from the methodto control resistance to flow and/or flow rate enhancement hereindescribed is capable of maintaining substantially constant flow rateenhancement. In some of those embodiments, the at least one framingassociative polymer has a weight-average molecular weight of 650,000g/mol to 750,000 g/mol and can be comprised at a concentration of about0.50*. In some of those embodiments the flow rate enhancement can beabout 28%.

In some embodiments, in the non-polar composition resulting from themethod to control resistance to flow and/or flow rate enhancement hereindescribed the flow rate enhancement is at least 20%. In some of thoseembodiments, the at least one framing associative polymer can have aweight-average molecular weight of 650,000 g/mol to 750,000 g/mol andcan be comprised at a concentration greater than 0.2c*.

In some embodiments, the non-polar composition resulting from the methodto control resistance to flow and/or flow rate enhancement hereindescribed is capable of maintaining a substantially constant flow rateenhancement in a pipeline of at least 8 kilometers. In some of thoseembodiments, the at least one framing associative polymer has aweight-average molecular weight greater than 650,000 g/mol and can becomprised at a concentration greater than 0.1c* possible 0.05c*. In someof those embodiments the composition can be in a flow having Reynoldsnumber equal to or higher than 5000, In some of those embodiments, ifminimization of shear degradation is desired the at least one framingassociative polymer can be provided at a weight-average molecular weight650,000 g/mol to 750,000 g/mol.

In some embodiments, the association constant of the at least oneframing associative polymer used in the method to control resistance toflow and/or flow rate enhancement of a non-polar composition is between7≤log₁₀ k≤14.

In some embodiments, the method to control resistance to flow and/orflow rate enhancement herein described can be applied to compositions ina flow having Reynolds number between about 5,000≤Re, and in particulargreater than 25,000 Re and the at least framing associative polymer asan association constant (k) in the range of 7≤log₁₀ k≤14.

In some embodiments the method to control resistance to flow and/or flowrate enhancement herein described can be applied to compositions in aflow having Reynolds number Re 25,000 and the at last one framingassociative polymer has an association constant (k) in the range of7≤log₁₀ k≤14.

In some embodiments, of the method to control resistance to flow and/orflow rate enhancement herein described, the concentration c is about 0.5c* or between about 0.5c* to 1c* and the another physical and/orchemical property is mist control.

In some embodiments, of the method to control resistance to flow and/orflow rate enhancement herein described, the concentration c is less thanapproximately c* and the another physical and/or chemical property isfuel efficiency.

In some embodiments, of the method to control resistance to flow and/orflow rate enhancement herein described, the concentration c is between0.1c* and 0.5c* and the another physical and/or chemical property isfuel efficiency.

In some embodiments, of the method to control resistance to flow and/orflow rate enhancement herein described, the concentration c is below orapproximately equal c* and the another physical and/or chemical propertyis enhanced lubrication.

In some embodiments, of the method to control resistance to flow and/orflow rate enhancement herein described, the concentration c is between0.05c* to c* and the another physical and/or chemical property isenhanced lubrication.

In some embodiments of the method to control resistance to flow and/orflow rate enhancement herein described, one or more capping associativepolymers having a weight-average molecular weight equal to or higherthan 200,000 g/mol can be comprised in an amount up to 20 wt % of atotal associative polymer concentration in the composition. In some ofthose embodiments, the one or more capping associative polymers can beprovided at 5 wt % of the total associative polymer concentration in thecomposition. In some of those embodiments, the one or more cappingassociative polymers can be provided at 10 wt % of the total associativepolymer concentration in the composition.

In some embodiments, framing associative polymer can be used, alone orin combination with capping associative polymers, in a method to controlsizes, and/or distribution of sizes, of the droplets of fluid (e.g. tocontrol fluid mist) in an associative non-polar composition in anassociative non-polar composition in a flow characterized by a Reynoldsnumber Re, and a characteristic length d. In some of those embodimentsone or more additional physical and/or chemical properties of theassociative non-polar composition can also be controlled. The methodcomprises providing a host composition having a viscosity μ_(h), adensity ρ_(h) and a dielectric constant equal to or less than about 5and providing at least one framing associative polymer substantiallysoluble in the host composition and having a weight-average molecularweight equal to or higher to 400,000 g/mol. The method further comprisescombining the host composition and the at least one framing associativepolymer herein described to provide the associative non-polarcomposition wherein the at least one framing associative polymer iscomprised at a concentration c selected between from about 0.05c* toabout 3c* based on the averaged molecular weight of the at least oneassociative polymer and on a physical and/or chemical property to becontrolled.

In In particular, in the method, the longest span of the at least oneframing associative polymer has a countour length ½L_(bf)≤L_(f)<L_(bf),wherein L_(bf) is a rupture length of the at least one framingassociative polymer in nanometers when the at least one framingassociative polymer is within the host non-polar composition at aconcentration c to provide the associative non-polar composition in aflow, L_(b) being given by implicit function

$F_{bf} = {\frac{{\pi\mu}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho\; d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$

in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³.

In some embodiments, the method can further comprise selecting aconcentration c of the at least one associative polymer in the hostcomposition, the concentration c selected between from about 0.05c* toabout 3c* depending on the averaged molecular weight and/or radius ofgyration of the at least one framing associative polymer and on aphysical and/or chemical property to be controlled based on the factorsherein described before the combining.

In the method herein described, when c≤2c*, μ is μ_(h), and ρ is ρ_(h),and when c>2c*, μ is the viscosity of the associative non-polarcomposition μ_(a), and ρ is the density of the associative non-polarcomposition μ_(a).

In some embodiments, the method can further comprise determining anoverlap concentration c* for the at least one associative polymer beforeperforming the selecting;

In some embodiments, in the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the at least oneframing associative polymer has a weight-average molecular weight equalto or higher than 1,000,000 g/mol, possible about 10,000,000 g/mol andcan be comprised at a concentration from 0.05 c* to 0.1 c*. in some ofthose embodiments, the at least one framing associative polymer providedin the has a weight-average molecular weight between 1,000,000 g/mol to4,000,000 g/mol, or preferably 2,000,000 g/mol to 4,000,000 g/mol, orbetween 1,000,000 and 2,000,000 g/mol if a longer lasting effect isdesired.

In some embodiments, in the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the at least oneframing associative polymer has a weight-average molecular weightbetween 400,000 g/mol to 1,000,000 g/mol and can be comprised at aconcentration between 0.5 c* and c*.

In some embodiments, in the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the at least oneframing associative polymer has a weight-average molecular weightbetween 400,000 g/mol to 1,000,000 g/mol, and can be comprised at aconcentration between c* and 3c*.

In some embodiments, the association constant of the at least oneframing associative polymer used in the method to control sizes, and/ordistribution of sizes, of the droplets of the fluid mist hereindescribed is between 7≤log₁₀ k≤14.

In some embodiments, of the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the concentrationc is about 0.5 c* or between about 0.5c* to 1c* and the another physicaland/or chemical property is drag reduction.

In some embodiments, of the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the concentrationc is less than approximately c* and the another physical and/or chemicalproperty is fuel efficiency.

In some embodiments, of the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the concentrationc is between 0.1c* and 0.5c* and the another physical and/or chemicalproperty is fuel efficiency.

In some embodiments, of the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the concentrationc is below or approximately equal c* and the another physical and/orchemical property is enhanced lubrication.

In some embodiments, of the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, the concentrationc is between 0.05c* to c* and the another physical and/or chemicalproperty is enhanced lubrication.

In some embodiments of the method to control sizes, and/or distributionof sizes, of the droplets of a fluid herein described, one or morecapping associative polymers having a weight-average molecular weightequal to or higher to 400,000 g/mol can be comprised in an amount up to20 wt % of a total associative polymer concentration in the composition.In some of those embodiments, the one or more capping associativepolymers can be provided in a 5 wt % of the total associative polymerconcentration in the composition. In some of those embodiments, the oneor more capping associative polymers can be provided in a 10 wt % of thetotal associative polymer concentration in the composition.

In some embodiments of the associative polymers, and relatedcompositions, methods and systems herein described any one of theassociative polymers herein described and in particular any one offraming associative polymers and/or capping associative polymers hereindescribed can have a weight-average molecular weight equal to or lowerthan 1,000,000 g/mol. In those embodiments, shear resistant associativepolymers can be provided. The wording “shear resistant” as used hereinin connection with a polymer indicates a polymer that, under amechanical stress sufficient to break a carbon-carbon covalent bond,shows a decrease in its weight-average molecular weight Mw equal to orlower than 5% and can be detected by techniques identifiable by askilled person. When a polymer is in a composition the mechanical stressapplied to different portions of the polymer are transmitted within thepolymer backbone and differently apply to different carbon-carboncovalent bonds of the chain based on the structure and configuration ofthe polymer as well as characteristics of flow as will be understood bya skilled person.

In embodiments where shear resistant associative polymers are desired,selection of one or more desired weight-average molecular weight can beperformed based on the structure of the backbone and presence, numberand location of secondary, tertiary and quaternary carbon atoms inbackbone as will be understood by a skilled person.

In some embodiments, framing associative polymers and/or cappingassociative polymers herein described can have a weight-averagemolecular weight the equal to or lower than 750,000 g/mol. In someembodiments, framing associative polymers and/or capping associativepolymers herein described can have a weight-average molecular weightbetween 400,000 g/mol and 1,000,000 g/mol. In particular in some ofthose embodiments shear resistant associative polymers can be a linearpolymer.

In some embodiments, shear resistant associative polymers hereindescribed can substantially maintain (±10%) control of one or morephysical and/or chemical properties in a non-polar composition afterapplication of a mechanical stress that is sufficient to break acarbon-carbon covalent bond (e.g. 150 kT where k is Boltzmann constant).For example such mechanical stress can be applied when a fluid passesthrough liquid handling operations, including pumping, turbulentpipeline flow, filters and the like as will be understood by a skilledperson. Accordingly, shear resistant associative polymers hereindescribed, and in particular shear resistant framing associativepolymers herein described can be used to provide non-polar compositionwhere a long lasting control of one or more properties is desired, andin particular where control of one or more desired effect is maintainedafter repeated exposure of the non-polar composition comprising theassociative polymer to the mechanical stress sufficient to break acarbon-carbon covalent bond.

In particular the mechanical stress sufficient to break a carbon-carboncovalent bond depends on various factors such as the chemical nature ofthe chain, the concentration and longest span of a polymer andadditional factors identifiable by a skilled person.

In particular in some embodiments, in which associative polymers hereindescribed are resistant to shear degradation (e.g. chain scission uponpassage through pumps, during prolonged turbulent flow in pipelines,tubes or hoses, during passage through filters), the associative polymerof the present disclosure can be introduced at early steps in thepreparation of non-polar host compositions. In many applications thehost composition can be itself a mixture.

In particular in exemplary embodiments in which a modified non polarcomposition comprising associative polymers herein described is providedin connection with production of inks or paints that can comprise acarrier liquid, pigments, stabilizers and other components, theassociative polymer can be added to the carrier liquid prior toincorporation of the remaining components, with the possibility that acentral depot of carrier liquid can feed production lines for diversecolors or grades of ink or paint. In some of these embodiments, theefficacy of the polymer can be retained after pumping, filtering, mixingand other processing steps.

Similarly, in exemplary embodiments in which a modified non polarcomposition comprising associative polymers herein described is providedin connection with lubricant applications, the associative polymersherein described can be incorporated into the base oil that issubsequently combined with diverse additive packages. At concentrationsup to c*, the associative polymers are expected to survive and areexpected to not interfere with processes that include but are notlimited to filtering, dewatering, pumping and mixing operations.

In exemplary embodiments in which a modified non polar compositioncomprising associative polymers herein described is provided inconnection with fuel applications (e.g. use as drag reducing agents,enhancers of fuel efficiency, emission reducing agents, or mist controlagents), the ability to incorporate the associative polymer hereindescribed at any point along the distribution system allows for exampleincorporation at the refinery; or in the intake line of a storage tank;or in the intake line of a tanker ship, railway tank car, tank of atanker truck; or in the intake line to a major site of use, such as anairport or a military depot; or in the transfer line from a storage tankinto a vehicle; or as a solution added to the tank of a vehicle at thetime of fueling.

In exemplary embodiments in which a modified non polar compositioncomprising associative polymers herein described is provided inconnection with drag reducing agents in the transport of petrochemicals(especially crude oil) through very long pipelines, the present polymersresist shear degradation upon passage through pumps; therefore, fewerinjection stations are required. In some cases, introduction of theassociative polymer at a single location prior to the intake of thepipeline will provide drag reduction throughout the entire length of thepipeline.

In some embodiments herein described associative polymers are notinterfacial agents, so that such polymers can be added prior todewatering operations (including but not limited to fuel handling) anddefoaming operations (including but not limited to production of paintsand inks); at concentrations up to c*, the associative polymers do notinterfere with these essential processing steps and the processing stepshave a minimal effect on the associative polymers.

In some embodiments, associative polymers herein described can be usedas a fuel additive with one or more of the following features: i)effective at low concentrations (acceptable viscosity), ii) introducedat the refinery; iii) resistant to non-intentional degradation; iv)soluble over wide temperature range (−50° C. to 50° C.); v) permitdewatering and filtering, vi) permit optimization in engine combustionchamber; vii) clean burning, and viii) affordable.

In some embodiments, the associative polymers and related compositionsherein described can be used in connection with application wherepassage of a fluid in a pipeline is performed. A turbulent drag, whichis usually expressed in terms of frictional pressure drop, plays acrucial role in pipeline transportation of non-polar liquids as will beunderstood by a skilled person: it increases the energy cost for movingthe liquid through the pipeline and thus limits the capacity of thesystem. Introducing a drag reducing agent (DRA) to the fluid, whichdampens turbulent regions near the pipe wall and consequently decreasesturbulent flow and increases laminar flow, provides a reduction in thefrictional pressure drop along the pipeline length. The benefitsprovided by DRAs include maintaining the same flow rate with asignificantly lower energy cost, and alternatively resulting in a muchhigher flow rate using the same amount of energy as will be understoodby a skilled person.

In some embodiments, the associative polymers here described can bedesigned to provide drag reduction to non-polar liquid in turbulentpipeline flow. In some of those embodiments when exposed to high shearflow in pump, aggregates of FGs serve as sacrificial weak links that canreversibly respond to the high shear by dissociation so as to protectthe backbone from degradation. Once the polymer chains leave the pump,they can re-form supramolecules via association of FGs and continue toprovide drag reduction to the pipeline flow. In some instancesidentifiable by a skilled person, associative polymers herein describedcan greatly simplify the practice of reducing energy cost for pipelinetransportation of non-polar hosts and/or increasing the capacity ofexisting pipeline system using drag reducing additives.

As disclosed herein, the associative polymers and non-polar compositionherein described can be provided as a part of systems to control atleast one rheological property of the drag reduction and/or flow rateenhancement alone or in combination with another physical and/orchemical properties herein described, including any of the methodsdescribed herein.

The systems can be provided in the form of kits of parts. In a kit ofparts, polymers (e.g. backbone polymers, associative polymers orprecursor thereof), compositions and other reagents to perform themethods can be comprised in the kit independently. One or more polymers,precursors, compositions and other reagents can be included in one ormore compositions alone or in mixtures identifiable by a skilled person.Each of the one or more polymers, precursors, compositions and otherreagents can be in a composition alone or together with a suitablevehicle.

Additional reagents can include molecules suitable to enhance reactions(e.g. association of one or more associative polymers herein describedwith a related host composition) according to any embodiments hereindescribed and/or molecules standards and/or equipment to facilitate orregulate the reaction (e.g. introduction of the associative polymer tothe host)

In particular, the components of the kit can be provided, with suitableinstructions and other necessary reagents, in order to perform themethods here described. The kit can contain the compositions in separatecontainers. Instructions, for example written or audio instructions, onpaper or electronic support such as tapes or CD-ROMs, for carrying outreactions according to embodiments herein described (e.g. introductionof associative polymer in a host composition), can also be included inthe kit. The kit can also contain, depending on the particular methodused, other packaged reagents and materials.

Further advantages and characteristics of the present disclosure willbecome more apparent hereinafter from the following detailed disclosureby way of illustration only with reference to an experimental section.

EXAMPLES

The associative polymers, materials, compositions, methods system hereindescribed are further illustrated in the following examples, which areprovided by way of illustration and are not intended to be limiting.

In particular, the following examples illustrate exemplary associativepolymers and related methods and systems. A person skilled in the artwill appreciate the applicability and the necessary modifications toadapt the features described in detail in the present section, toadditional associative polymers, compositions, methods and systemsaccording to embodiments of the present disclosure.

Example 1: Exemplary Associative Polymer and Architectures

Exemplary associative polymers and related exemplary architectures areillustrated in FIGS. 3 to 6.

In particular in the illustration of FIG. 3 a linear polymer backbone of1,4-polybutadiene is illustrated in which end groups are <1 wt % of thepolymer and contain <0.2 wt % heteroatoms. When added to fuel, polymersof this type burn cleanly and maintain the caloric content of the fuel.

The illustration of FIG. 4 provides exemplary functional groups whichcan be used with the backbone of FIG. 3 or other backbones as will beunderstood by a skilled person. The illustration of FIGS. 5 and 6 showsexemplary branched architectures (FIG. 5) and exemplary block-polymerarchitecture (FIG. 6) which can be created with the backbone of and/orother backbones as will be understood by a skilled person. When theassociative polymer is added to a host composition the FGs form physicalassociations according to their nature (e.g. self to self,donor-acceptor, pairwise, or multidentate). The illustration of FIGS. 1and 2 show exemplary types of supramolecular structures thus formed.

Example 2: Methods of Making Associative Polymers and RelatedArchitectures

A schematic illustration of exemplary reactions and methods suitable tomake associative polymers herein described is provided in FIGS. 7 to 10.

In particular FIG. 7 shows a schematic of an exemplary method to providean associative polymer herein described illustrated making specificreference to embodiments where a corresponding non-polar composition isa fuel.

FIGS. 8 and 9 show an exemplary ROMP+Chain Transfer Agent (CTA) reaction(FIG. 8) and exemplary chain transfer agents (FIG. 9). This exemplaryreaction allows in several cases precise control of the number ofassociating groups. It will be appreciated by a skilled person that itcan be straightforward to synthesize and purify at large scaleassociative polymers compatible with non-polar compositions, with thebackbone and associative groups chosen for a particular application asdescribed in the specification (see, e.g., [27-29]).

FIG. 10 shows a schematic of an exemplary method to synthesize anassociative polymer using CTAs.

Example 3: Synthesis of High Molecular Weight Di-TE PB by ROMP

[00389] 6.7 mg of octa-functional tert-butyl ester CTA is loaded into a50 ml Schlenk flask (charged with a magnetic stir bar). The flask islater sealed with a septum. The content is then deoxygenated by 5 timesof pulling vacuum/filling argon. 0.5 ml of deoxygenated DCM is added todissolve the CTA. 0.13 ml of 1 mg/ml DCM solution of Grubbs II catalystis injected into the flask, and then 0.03 ml of freshly vacuumdistilled, purified COD (≡50 eq. w.r.t. CTA) is immediately injected.

The mixture is stirred at 40° C. for 33 minutes to allow completeincorporation of CTA into the polymer. Another 0.13 ml of freshlyprepared 1 mg/ml DCM solution of Grubbs II catalyst is then injected,followed by 5.6 ml of freshly vacuum distilled, purified COD (≡10,000eq.) in 12 ml of deoxygenated DCM. The reaction is stopped by adding 30ml of oxygen-containing DCM as the mixture turns viscous enough tocompletely stop the motion of magnetic stir bar. The diluted mixture isprecipitated into 400 ml of acetone at room temperature. The resultingpolymer is collected and dried in vacuo at room temperature overnight.GPC results of the polymer: M_(w)=430,000 g/mol, PDI=1.46.

Example 4: Deprotection of the Acid End Groups

1 g of the aforementioned polymer is loaded into a 50 ml Schlenk flask(charged with a magnetic stir bar), and degassed by 5 times of pullingvacuum/filling argon). 30 ml of deoxygenated is then syringe-transferredinto the flask. The mixture is homogenized at room temperature. Oncecomplete homogenization is achieved, 1.25 ml of deoxygenatedtrifluoroacetic acid (TFA) is syringe-transferred into the flask. Themixture is then stirred at room temperature overnight.

Upon the completion of TFA hydrolysis, the mixture is diluted with 20 mlof DCM, and the resulting solution is precipitated into 400 ml ofacetone at room temperature. The resulting polymer is further purifiedby 2 times of re-precipitation from THE into acetone.

Example 5: Synthesis of High Molecular Weight Di-TB PB by ROMP

Synthesis of high M.W di-TB PB by ROMP is performed according to thefollowing steps:

Step 1: Prepolymer Synthesis

5 mg of octa-functional chloro CTA is loaded into a 50 ml Schlenk flask(charged with a magnetic stir bar). The flask is later sealed with aseptum. The content is then deoxygenated by 5 times of pullingvacuum/filling argon. 0.5 ml of deoxygenated DCM is added to dissolvethe CTA. 0.13 ml of 1 mg/ml DCM solution of Grubbs II catalyst isinjected into the flask, and then 0.03 ml of freshly vacuum distilled,purified COD (≡50 eq. w.r.t. CTA) is immediately injected. The mixtureis stirred at 40° C. for 33 minutes to allow complete incorporation ofCTA into the polymer. Another 0.13 ml of freshly prepared 1 mg/ml DCMsolution of Grubbs II catalyst is then injected, followed by 5.6 ml offreshly vacuum distilled, purified COD (≡10,000 eq.) in 12 ml ofdeoxygenated DCM. The reaction is stopped by adding 30 ml ofoxygen-containing DCM as the mixture turns viscous enough to completelystop the motion of magnetic stir bar. The diluted mixture is thenprecipitated into 400 ml of acetone at room temperature. The resultingpolymer is collected and dried in vacuo at room temperature overnight.GPC results of the polymer: M_(w)=430,000 g/mol, PDI=1.46.

Step 2: End-Azidation of Prepolymer

1 g of the aforementioned chloro-terminated prepolymer is loaded into a50 ml Schlenk flask, and dissolved into 30 ml of anhydrous THF. Uponcomplete homogenization, 0.73 g of azidotrimethylsilane (≡1200 eq w.r.t.polymer) and 1.57 g of tetrabutylammonium fluoride (≡1200 eq w.r.t.polymer) are added into the flask. The resulting mixture is degassed by2 freeze-pump-thaw cycles to prevent crosslinking by dissolved oxygen.Then, the mixture is stirred at 60° C. overnight. The mixture isprecipitated into 300 ml of methanol at room temperature. The resultingpolymer is further purified by 2 more times of reprecipitation from THEinto acetone. The resulting polymer is dried in vacuo at roomtemperature overnight.

Step 3: Attachment of Tertiary Amine Groups to Polymer Chain Ends

0.68 g of the aforementioned azido-terminated prepolymer is loaded intoa 50 ml Schlenk flask, and dissolved into 25 ml of anhydrous THIF. Oncehomogenization is complete, 0.23 g of 3-Dimethylamino-1-propyne (≡1,200eq. w.r.t. the polymer), along with 0.02 g ofN,N,N′,N′,N″-pentamethyldiethylenetriamine (PMDETA, ≡50 eq. w.r.t. thepolymer) are added into the flask. The mixture is then deoxygenated by 2freeze-pump-thaw cycles. Later it is frozen and pumped again, and then0.016 g of copper (I) bromide (≡50 eq. w.r.t. the polymer)) is addedinto the flask under the protection of argon flow when the mixture isstill frozen. After thawing the mixture and filling the flask withargon, the mixture is stirred at room temperature for 20 minutes inorder to homogenize the copper (I) catalyst. The mixture is stirred at50° C. overnight. 2 ml of methanol is slowly injected into the mixturein order to remove copper from the amine end groups. The mixture isprecipitated into 300 ml of methanol at room temperature. The resultingpolymer is further purified by 2 more times of reprecipitation from THEinto methanol. It is later dried in vacuo at room temperature overnight.

Example 6: Effect of Self-Association in Exemplary Associative Polymers

Proof of effect of self-association in exemplary associative polymersherein described is illustrated in FIG. 11 and FIG. 12. In the exemplaryassociative of Example 5 the aforementioned method of recovering the endacid groups does not crosslink the polybutadiene backbone, as proved inthe superposition of GPC traces of 430K di-TE PB and the resultingpolymer of its hydrolysis reaction (in TIF) illustrated in FIG. 11

In the illustration of FIG. 11, the slight increase in the population ofhigh molecular weight species is due to the weak self-association ofchain-end acid clusters. The apparent M_(w) increases by 20% after TFAhydrolysis.

A further confirmation is provided by the illustration of FIG. 12. Inparticular, FIG. 12 shows the rheology data of the 1 wt % Jet-Asolutions of the 430K di-TE PB and 430K di-TA PB respectively. Theviscosities of 1 wt % Jet-A solution of 430K di-TA PB are significantlyhigher than those of the ester prepolymer. Since the GPC results showthe extent of backbone crosslinking during removal of tert-butyl groupsis negligible, it is reasonable to say that the self-association of acidclusters accounts for the increase in viscosities.

Example 7: Effect of End-to-End Donor Acceptor Association in ExemplaryAssociative Polymers

A proof of the effect of end-to-end donor/acceptor association isprovided in FIG. 13 and FIG. 14. In particular FIG. 13, shows thesuperposition of GPC traces of the 430K octa chloro PB and thecorresponding octa tertiary amine PB.

In the illustration of FIG. 13, the polybutadiene backbone is mainlyintact after two end-functionalization reactions.

FIG. 14 shows the rheology data of 1 wt % Jet-A solutions of 430K di-TEPB, di-TA PB, di-TB PB, and 1:1 w/w di-TA PB/di-TB PB mixture. In theillustration of FIG. 14, the 1:1 mixture shows significantly higherviscosities than the other solutions. Since none of the two polymercomponents are crosslinked, it suggests that the end-to-end acid/baseinteraction results in the formation of supramolecular species.

Example 8: Effect of an Exemplary Associative Polymer on FuelCompositions

Effect of di-TA PB synthesized according to Example 5, was tested in JetA fuel. In particular a composition comprising 0.5% of di-TA PB with abackbone length of 264,000 g/mol (denoted 264K di-TA PB) in jet A fuelhas been provided as illustrated in FIG. 15.

In the illustration of FIG. 15 is shown that the exemplary associativedi-TA PB of Example 5 showed no phase separation and was able to stay insolution (crystal clear) even at −30° C. for months (see FIG. 15, PanelA).

Additionally, dewatering operations appeared to occur as quickly andcompletely in the composition with associative di-TA PB of Example 5, asin the untreated host Jet A (see FIG. 15, Panel B left vial v. rightvial).

Example 9: High-Speed Impact/Flammability Test

To demonstrate the effect of exemplary polymers on the mist-control ofkerosene, a series of high-speed impact/flammability test were conductedat California Institute of Technology. The high-speed impact test isdesigned to simulate a scenario in which fuels can be atomized intodroplets due to impact, whereas the continuously provided ignitionsources are used to obtain an indication of the flammability ofresulting droplets. The following samples were loaded into 50 mlaluminum cans, fixed on a stage, and impacted by a 5 cm×3 cm steelcylinder travelling at 200 km/hr (three continuously burning propanetorches were set up along the path of splashed samples): Jet-A, 0.35 wt% Jet-A solutions of 4.2 M polyisobutylene (PIB) with and withoutrecirculation by a Bosch 69100 In-line turbine fuel pump for 1 minutes,0.3 wt % of Jet-A solutions of 430K di-TA PB with and withoutrecirculation by a Bosch 69100 In-line turbine fuel pump for 1 minutes.The results for each sample are described below: Jet-A: Significantamount of fine droplets was generated upon impact. The fine dropletstravelling along the path of the projectile were ignited by the burningtorches within 50 milliseconds, and then evolved into a propagating fireball.

0.35 wt % Jet-A solution of 4.2M PIB, without shear: Large droplets andfilaments were generated by the impact. Sparkles were observed as thefluid elements passed over the torches, but they failed to propagate.

0.35 wt % Jet-A solution of 4.2MPIB, with 1 min. of shear: Fine dropletswere generated by the impact. The fine droplets travelling along thepath of the projectile were ignited by the burning torches within 50milliseconds, and then evolved into a propagating fire ball.

0.3 wt % Jet-A solution of 430K di-TA PB, without shear: Droplets weregenerated by the impact. Sparkles were observed as the fluid elementspassed over the torches, but they failed to propagate.

0.3 wt % Jet-A solution of 430K di-TA PB, with 1 min. of shear: Dropletswere generated by the impact. Sparkles were observed as the fluidelements passed over the torches, but they failed to propagate.

Example 10: Synthesis of Octa Functional CTAs

Reaction schemes for exemplary Octa functional CTAs in accordance withthe present disclosure are shown in the illustration of FIG. 18 and FIG.19.

Example 11: Exemplary Node to Chain and Node to FG Interactions

Exemplary pairs of reactive groups that are useful at end positions suchas R₁ or R₂ in the structure of formula (III) or in di- or multi-valentcrosslinkers and the product of their reaction, which can be used forcovalently linking a chain and a FG, or linking chains to a node orattaching FG to a node in accordance with the present disclosure, areshown in the illustration of FIG. 20 and FIG. 21.

Example 12: Polymer-Composition Solubility Determination

Solubility of an exemplary polymer 1,4-polybutadiene (PB) in a non-polarcomposition has been determined. The nonpolar composition is kerosene,which can be considered to be a mixture of hydrocarbons that contain6-16 carbon atoms per molecule, the v₀ of octane (160 cm³/mol) can bechosen as a representative value for kerosene.

Accordingly, when 1,4-polybutadiene (PB) is used as the backbone ofinvented associative polymers, the value of δ₁ is ˜8 (cal/cm³)^(0.5)(see, e.g. [18, 33]). To evaluate 62 for kerosene, the followingrelationship (dispersive Hansen parameter) can be used:

δ=9.55n _(D)−5.55

where n_(D) is the refractive index of the host, and n_(D) can bewell-approximated by the square root of the dielectric constant (ε) ofthe host. Given ε_(kerosene) is 1.8 at 20° C., δ₂ is˜9.55×(1.8)^(0.5)−5.55=7.26.

Accordingly, the interaction parameter for the associative polymer witha 1,4-polybutadiene backbone in kerosene at ambient temperature can beestimated as follows:

${{\chi \approx {{{0.3}4} + {\frac{160}{1.987 \times 29{8.1}5} \times \left( {8 - {{7.2}6}} \right)^{2}}}} = {{0.4}9}}.$

The calculated value of χ of 0.49 indicates that the PB associativepolymer with a 1,4-polybutadiene backbone would be expected to besubstantially soluble in a non-polar composition of kerosene.

A skilled person can determine based on the above Example if otherassociative polymer backbones would be substantially soluble in othernon-polar compositions by applying the same calculations using theparticular solubility parameters for the particular non-polarcomposition.

Example 13A: Drag Reduction Test

0.2 grams of telechelic 1,4-PB of M_(w) 630,000 g/mol, terminated by 2acid groups (denoted 630K di-DA PB) and 0.2 grams of telechelic 1,4-PBof M_(w) 540,000 g/mol, terminated by 2 tertiary amine groups (denoted540K di-DB PB) were dissolved in 39.6 grams of Jet-A at room temperatureover 16 hours.

The resulting 1 wt % Jet-A solution of 1:1 w/w 630K di-DA PB/540K di-DBPB was further diluted with 1293 grams of Jet-A to a concentration of300 ppm (˜0.1c* of the non-associative backbone). A Bosch 69100 In-lineturbine fuel pump with its outlet connected to a piece of TYGON® tubing(inner diameter=6.34 mm; length=40 cm) and inlet outlet connected to apiece of TYGON® tubing (inner diameter=3.17 mm; length=2.14 m) was usedto transfer the fuel sample from its reservoir to a collecting jar overa period of 20 seconds (FIG. 39A).

The pump was primed with ˜200 mL of the sample before the test. Thecollecting jar was weighed before and after the transfer in order todetermine the amount of fuel collected. The same procedure was alsoperformed on the unmodified host Jet-A. The measured mass flow rate ofunmodified Jet-A was 24.17 g/s, which corresponded to a Reynolds numberof 6458. As for the Jet-A sample with 300 ppm of 1:1 donor/acceptorpolymer pair, the measured mass flow rate was 24.92 g/s. Hence, anincrease of 3.2% in mass flow rate was achieved, indicating that thepresence of 1:1 (w/w) mixture of 630K di-DA PB and 540K di-DB PB at 300ppm in Jet-A reduced the effect of turbulent drag on flow rate.

A skilled person will realize that the above test can be applied toother associative polymers in order to determine the extent of dragreduction.

Example 13B: Long Lasting Drag Reduction Test

0.7 grams of telechelic 1,4-PB of M_(w) 670,000 g/mol, terminated by twoacid groups (denoted 670K di-DA PB) and 0.7 grams of telechelic 1,4-PBof M_(w) 630,000 g/mol, terminated by 2 tertiary amine groups (denoted630K di-DB PB) were dissolved in 139 grams of Jet-A at room temperatureover 16 hours. The resulting 1 wt % Jet-A solution of 1:1 w/w 670K di-DAPB/630K di-DB PB was further diluted with 1133 grams of Jet-A to aconcentration of 1,100 ppm (˜0.37c* of the nonassociative backbone).

3.2 grams of polyisobutylene of M_(w) 4,200,000 g/mol (denoted 4.2M PIB)were dissolved in 637 grams of Jet-A at room temperature over 48 hours.52 grams of the resulting 0.5 wt % Jet-A solution was further dilutedwith 1133 grams of Jet-A to a concentration of 217 ppm.

The apparatus for drag reduction study is shown in FIG. 39B. A2.5-gallon cylindrical steel air tank (Viair 91208, 200 psi rated) wasused as a pressurizable sample reservoir, which was fitted with apressure gauge, a high-pressure gas inlet, a 200-psi safety reliefvalve, and a ball valve as the sample outlet. A 10-liter polyethylene(PE) bottle with a tubulation connector at the bottom was used as thesample receiving container. A 9.15-meter long piece of PTFE tubing(I.D.=3.17 mm) connecting the outlet valve of the air tank and thetubulation connector of the PE bottle was used as a miniature pipelinewhere turbulent drag took place.

Test samples include untreated Jet-A as the reference, Jet-A solution of4.2M PIB at 217 ppm as the control, and Jet-A solution of 1:1 (w/w) 670KDi-DA PB/630K Di-DB PB at 1,100 ppm. Gravity flow was used to transferthe test sample from the 10-liter PE bottle into the air tank over aperiod of 35 min. The test fluid was pressurized to 200 psi by means ofhigh-pressure nitrogen. Flow rates were determined via a catch-and-weightechnique: The test fluid was driven through the PTFE tubing over aperiod of 21 s to the 10-liter PE bottle, which was weighed before andafter the test to determine the average mass flow rate and thecorresponding Reynolds number (Re). Five passes were performed on eachsample. When untreated Jet-A was tested, a Re of 10,770 was achieved,which indicates the apparatus was able to generate turbulent flow.Results are expressed as % flow enhancement defined as 100* (Jet-Asolution flow rate—Jet-A flow rate)/(Jet-A flow rate) with all flowrates compared at common initial pressure (200 psi) and final pressure(192 psi). The results are shown in FIG. 39C and FIG. 68.

Compared to untreated Jet-A, the presence of 4.2 M PIB at 217 ppm inJet-A initially helped improve the flow rate by 37.4%. The flow rateenhancement steadily decreased as the number of passes through thesystem increased: at the 5th pass, the flow rate enhancement by 4.2M PIBat 217 ppm in Jet-A was reduced to 26.2%, indicating shear degradationof 4.2 M PIB in turbulent flow.

The flow rate enhancement by 1:1 (w/w) 670K Di-DA PB/630K Di-DB PB at1,100 ppm in Jet-A was found long-lasting throughout the five passes: nomeasurable decrease in flow rate enhancement was observed(average=28.2%, standard deviation=0.27%). The results show that at1,100 ppm in Jet-A, 1:1 (w/w) 670K Di-DA PB/630K Di-DB PB resist sheardegradation in turbulent flow and thus provide long-lasting dragreduction thus supporting the conclusion that the flow rate enhancementcan be maintained constant with flow having a high Reynolds number (e.g.higher than 5000 or 25000) and/or along a long pipeline (e.g. 8 Km ormore)

Example 14: Detection of Rehological Properties of Solutions

The methods presented in Examples 2-5 to synthesize telechelic 1,4-PBswith M_(w) up to 430,000 g/mol capped at each end with well-definedtert-butyl ester-terminated dendrons (FIG. 41) provides facile access tomatched pairs of non-associative and associative telechelic 1,4-PBs(FIG. 42). In this example, these model polymers were used to study therelationship between molecular properties (e.g., polymer molecularweight and the number of carboxyl groups on chain ends) and associationbehavior, particularly its effects on the rheological properties insolution. The present study of the self-association behavior ofcarboxyl-terminated telechelic 1,4-PBs provides a foundation forcomparative studies of complementary association illustrated in FIG. 22.

The following materials and methods were used: Solvents 1-chlorododecane(CDD) and tetralin (TL) were both obtained from Aldrich in 97% and 99%purity, respectively. All tert-butyl ester-terminated telechelic 1,4-PBsand their corresponding carboxyl-terminated telechelic 1,4-PBs wereprepared as described herein. Four values of the number of functionalgroups on polymer chain ends, N, and three polymer backbone lengths (interms of M_(w) by GPC-LS) were selected for the present study: A seriesof polymers with approximately matched backbone length (nominally220,000 g/mol) were prepared with N=1, 2, 4 and 8; and a series ofpolymers with N=4 was prepared with three backbone lengths of 76,000,230,000, and 430,000 g/mol. (Table 3.1). To simplify the nomenclature ofmaterials, polymer end-groups with N=1, 2, 4, and 8 tert-butyl estergroups are denoted ME, DE, TE, and OE (for mono-, di-, tetra-,octa-ester end groups, respectively), respectively. Similarly, polymerend-groups with N=1, 2, 4, and 8 carboxyl groups are denoted MA, DA, TA,and OA (for mono-, di-, tetra-, octa-acid end groups, respectively),respectively

Procedure for Sample Preparation: Solutions of tert-butyl esterterminated polymers for viscosity measurements were prepared bycombining polymer and solvent in clean 20 mL scintillation vials orlarger 50 mL glass jars which were placed on a Wrist-Action Shaker(Burrell Scientific) for up to 24 h to allow complete homogenization.

Solutions of carboxyl-terminated polymers were prepared as follows: To150 to 200 mg of carboxyl-terminated polymer in a 50-mL Schlenk flaskwas added necessary amount of solvent for 1 wt % stock. The contents ofthe Schlenk flask were degassed by 3 freeze-pump-thaw cycles, and thenstirred overnight at 70° C.

Viscosity Measurements: Steady shear viscosity was measured in acone-plate geometry (60 mm diameter aluminum, 1° cone, 29 m truncation)using an AR1000 rheometer from TA Instruments (temperature controlled at25° C.). Solutions of tert-butyl ester terminated polymers were probedin the shear rate range 1-200 s⁻¹ logarithmically (5 shear rates perdecade). The range was extended to 3000 s⁻¹ for carboxyl-terminatedpolymers to better capture shear-thinning behavior. All viscosity datawere reported in terms of specific viscosity(η_(sp),≡(η_(solution)−η_(solvent))/η_(solvent), where η_(solvent)=2.72mPa·s for CDD and 2.02 mPa·s for TL at 25° C.) which reflects thecontribution of the polymer to the viscosity [34].

Example 15: Dissolution Behavior

All six tert-butyl ester-terminated 1,4-PBs (Table 7) were found readilysoluble in both CDD and TL. With increasing carboxyl content, it becamemore difficult to dissolve carboxyl-terminated polymers: For N=1, thecorresponding polymer (226K di-MA 1,4-PB) was found soluble in both CDDand TL at room temperature; at N=2 and 4, the corresponding polymers(230K di-DA 1,4-PB; 76K, 230K, and 430K di-TA 1,4-PBs) were not solublein either model solvent at room temperature, but they dissolved into CDDand TL when heated at 70° C. and remained in solution thereafter. AtN=8, the polymer 207K di-OA 1,4-PB did not dissolve completely intoeither solvent even when heated at elevated temperatures (>110° C.)overnight. The difficulty of dissolving 207K di-OA 1,4-PB is not due tocrosslinking: The polymer dissolves readily in THF, it passes easilythrough filters, and GPC-LS analysis showed that 207K di-OA 1,4-PB has aunimodal distribution similar to the other polymers in the series ofsimilar M_(w) (near 220,000 g/mol; see Table 7, which shows molecularweight (M_(w)) and number of chain-end functional groups (N) oftert-butyl ester- and carboxyl-terminated telechelic 1,4-PBs).

TABLE 7^(a) Nominal M_(w) N 76 220 430 1 226 (1.4) 2 230 (1.5) 4 76(1.5) 230 (1.4) 430 (1.5) 8 207 (1.5) ^(a)GPC was performed for in THFfor 35° C. for the tert-butyl ester form; results are shown for M_(w) inkg/mol followed by PDI in parentheses.

Example 16: Steady-Flow Shear Viscosity of 1 wt % Polymer Solutions

Specific viscosity (η_(sp)) of 1 wt % polymer solutions averaged overshear rates from 10-100 s⁻¹ show that all solutions ofcarboxyl-terminated 1,4-PBs had higher η_(sp) than their tert-butylester-terminated (i.e., protected) counterparts, but the highestincrease was observed in the case of N=4 (FIG. 23). The lack of η_(sp)data for carboxyl-terminated 1,4-PB with N=8 is due to the poorsolubility of the polymer in both solvents. While η_(sp) for all of thenon-associative ˜230K tert-butyl ester-terminated polymers was the same,the deprotection of carboxyl groups on polymer chain ends produced athreefold increase in specific viscosity in both CDD and tetralin forN=4, whereas at N=1 and 2 only marginal increases were observed afterdeprotection of carboxyl groups (FIG. 23). Thus, there appears to be aminimum number of carboxyl groups on polymer chain ends to achieve theintermolecular association suitable for viscosity modification (N>2) anda maximum number imposed by the solubility limit (N<8). The effect ofsolvent quality on η_(sp) was also observed in FIG. 23. Increasing thelength of 1,4-PB backbone, for identical TA end groups (N=4) increasesthe specific viscosity strongly (FIG. 23): In tetralin, for the 76,000g/mol polymer, deprotection of carboxyl groups only increases thespecific viscosity by 90%, whereas the increase is more than 320% forthe 430,000 g/mol polymer. For each polymer, η_(sp) of its 1 wt %tetralin solution was found nearly twice as high as that of its 1 wt %1-chlorododecane solution.

Example 17: Concentration Dependence of Specific Viscosity

While the values of η_(sp) of three tert-butyl ester-terminated polymersin both CDD and TL showed a nearly linear dependence on polymerconcentration, the CDD and TL solutions of the three carboxyl-terminatedpolymers (76K, 230K and 430K di-TA 1,4-PBs) exhibited nonlinearincreases of η_(sp) with concentration, and the extent of suchnon-linearity was found positively correlated with the M_(w) of polymerbackbone (FIG. 24). In accord with the observation that associativepolymers with 1 and 2 carboxyl groups at their ends have less effect onviscosity, comparison of the three 230K carboxyl-terminated 1,4-PBs withN=1, 2 and 4 shows that the non-linear increase of η_(sp) with polymerconcentration was obvious only in the case of N=4 (FIG. 25).

Example 18: Shear-Thinning Behavior of Solutions of Carboxyl-TerminatedPolymers

The onset and magnitude of shear-thinning can depend on the molecularweight and concentration of polymer. Solutions of 76K di-TA 1,4-PBshowed negligible shear-thinning (up to 3000 s⁻¹) (in either CDD or TL,FIGS. 33 and 34, respectively). In the case of 230K di-TA 1,4-PB, itsCDD and TL solutions showed shear-thinning at 1 wt %, with onsets in therange 10-100 s⁻¹. With decreasing concentration, the magnitude of shearthinning decreased and the shear rate required to elicit it increased(e.g., relative to the 1 wt % solution, at 0.7 wt %, the extent ofshear-thinning observed in both CDD and TL was less significant and theonset shifted to >100 s⁻¹) (FIGS. 33 and 34). Similar trends wereobserved for solutions of 430K di-TA 1,4-PBs, with greater extent ofshear-thinning and onset of shear-thinning at lower shear rates comparedto their 76K and 230K counterparts (in both CDD and TL, FIGS. 33 and 34,respectively).

An interesting shear-thickening feature followed by furthershear-thinning was observed for 430K di-TA 1,4-PB at 1 wt % in CDD and0.7 wt % in TL (see FIGS. 26A and 26B). The shear-thickening appeared ata higher shear rate in CDD than in TL (shear rates between 250 and 1000s⁻¹ in FIG. 26, compared to 160 and 630 s⁻¹ in FIG. 26B).

Example 19: 1H NMR Study on Complementary End-Association in DeuteratedChloroform

¹H NMR spectroscopy has been widely used to study the association ofhydrogen-bonding-based hetero-complementary associative motifs innon-polar deuterated solvents (e.g., CDCl₃) because the resultanthydrogen bonds can cause significant changes in electron environmentssurrounding protons participating complementary associations;consequently, measurable changes in chemical shifts of those protons canbe observed as the results of such complementary associations [35-41].This technique was adopted to investigate if the three pairs ofhetero-complementary associative groups (THY/DAAP, HR/CA, and TA/TB) canperform complementary association in CDCl₃ at room temperature whenattached to chain ends of 1,4-PB of M_(w)˜10,000-50,000 g/mol, which waschosen to keep signals of end-groups recognizable.

¹H NAMR Study of Hetero-Complementary End-Association. ¹H NMR study ofhetero-complementary end-association of telechelic 1,4-PB chains wascarried out at a total polymer concentration of ˜1 wt % in deuteratedchloroform (CDCl₃) at room temperature. ¹H NMR samples of individualtelechelic associative polymers were prepared by combining polymer andCDCl₃ at a polymer concentration ˜1 wt % in 20 mL scintillation vials,which were placed on a Wrist-Action Shaker (Burrell Scientific) for upto 16 h to allow the polymer to completely dissolve. ¹H NMR samples ofcomplementary polymer pairs were prepared by mixing ˜1 wt % CDCl₃solutions of their corresponding polymers in 20 mL scintillation vialsin desired end-group ratios, except for the 1:1 (w/w) mixture of 24Kdi-TA/22K di-TB 1,4-PBs, of which the ¹H NMR sample was prepared bycombining the two polymers at a 1:1 weight ratio and CDCl₃ at a totalpolymer concentration˜1 wt % in a 20 mL scintillation vial that wasplaced on a Wrist-Action Shaker (Burrell Scientific) for 16 h at roomtemperature.

The investigation of hetero-complementary end-association by ¹H NMRspectroscopy was carried out by measuring the ¹H NMR spectra ofindividual telechelic associative polymers and those of complementarypolymer pairs, followed by comparing signals of protons participatinghetero-complementary end-association in ¹H NMR spectra of individualpolymer solutions to those of the same protons in the spectra ofcorresponding polymer mixtures. Due to the inherent detection limit of1H NMR spectroscopy, either changes in chemical shifts or thedisappearance of the signals of protons participatinghetero-complementary association of polymer end-groups were followed asthe evidence of end-association, depending on the sizes of polymerbackbones. For telechelic associative polymers of M_(w)≤50,000 g/mol,characteristic shifts of signals of associative end-groups werefollowed; for those of M_(w)≥200,000 g/mol, the focus was whether themixing of complementary partners caused the disappearance of the signalsof protons participating hetero-complementary association of polymerend-groups.

¹H NMR spectra were obtained using a Varian Inova 500 spectrometer (500MHz); all spectra were recorded in CDCl₃, acetone-d6, and DMSO-d6 atambient temperature. Chemical shifts were reported in parts per million(ppm, 6) and were referenced to residual solvent resonances. Polymermolecular weight measurements were carried out in tetrahydrofuran (THF)at 35° C. eluting at 0.9 mL/min (pump: Shimadzu LC-20AD Prominence HPLCPump) through four PLgel 10-μm analytical columns (Polymer Labs, 10⁶ to103 Ain pore size) connected in series to a DAWN EOS multi-angle laserlight scattering (MALLS) detector (Wyatt Technology, Ar laser, λ=690 nm)and a Waters 410 differential refractometer detector (λ=930 nm).

The results of each pair are described as follows:

THY (thymine) DAAP (diacetamidopyridine): FIG. 27 shows the expanded ¹HNMR spectra (500 MHz, CDCl₃) of 10K di-THY 1,4-PB 5, 10K di-DAAP 1,4-PB14, and the mixture of 5 and 14 in a 1:2 wt ratio. In the absence of itscomplementary unit, the signal of the imide proton of THY end groups wasobserved at 8.05 ppm (FIG. 27). Upon addition of ˜2 eq of DAAP endgroups, a large downfield shift to 11.05 ppm accompanied by signalbroadening was observed (FIG. 27). Similar shift was also observed forthe signal of the amide protons of DAAP end groups (from 7.58 to 8.42ppm, in FIG. 27, panels B and C). The observed association-induced shift(˜2.9 ppm) of the imide proton signal of THY end groups is in goodagreement with the literature [36, 37, 39], and it indicates that THYand DAAP end groups could find and associate with each other in CDCl₃.

HR (Hamilton receptor) CA (cyanuric acid): FIG. 28 shows the expanded ¹HNMR spectra (500 MHz, CDCl₃) of 50K di-CA 1,4-PB, 24K di-HR 1,4-PB, andthe mixture of 50K di-CA 1,4-PB and 24K di-HR 1,4-PB in a 1:1.4 wtratio. In the absence of its complementary unit, the signal of the imideprotons of the CA end group was observed at 7.75 ppm (FIG. 28). A verylarge downfield shift to 12.90 ppm accompanied by peak broadening wasobserved (FIG. 28) as 2 eq of HR end groups were added. Similar to thecase of THY/DAAP pair, the observed association-induced shift (˜5.2 ppm)of the signal of the imide protons of CA units indicates that CA and HRend groups could also find and associate with each other in CDCl₃. Themagnitude of the observed shift is in good agreement with the literature[42-47].

TA TB: Due to the fact that 24K di-TA 1,4-PB is not soluble in CDCl₃, ¹HNMR study was only performed on 22K di-TB 1,4-PB and its 1:1 (w/w)mixture with 24K di-TA 1,4-PB and monitored the association by trackingthe shifts of the signals of the tertiary amine end group (H₁ and H₂,see FIG. 29). The results are shown in FIG. 29. It was found that thepresence of 22K di-TB 1,4-PB assisted the dissolution of 24K di-TA1,4-PB in CDCl₃ and thus rendered the ¹H NMR experiment possible. Thesignals of H₁ and H₂ were observed at 2.28 and 3.60 ppm respectively inthe absence of 24K di-TA 1,4-PB (FIG. 29). The addition of 24K di-TA1,4-PB resulted in shifts of both signals: The signals of H₁ and H₂shifted from 2.28 and 3.60 to 2.46 and 3.85 ppm, respectively. Theobserved shifts indicate the association of TA and TB end groups.

In order to determine if the three pairs of complementary associativegroups were still effective when attached to chain ends of 1,4-PBs ofM_(w)˜200,000-300,000 g/mol, ¹H NMR analysis of the correspondingpolymers and the complementary pairs was performed at ˜1 wt % in CDCl₃at room temperature. It was found that in this case, signals of polymerend groups were barely recognizable due to their low contents in thetest samples. In addition, association-induced signal broadening couldcause signals of protons involved in complementary association to appearvanished. Nevertheless, evidence of end-association was observed in allthree pairs of telechelic associative polymers of M_(w)˜200,000 g/mol.In the case of the THY/DAAP pair, the signal of the imide proton of THYend group of 288K di-THY 1,4-PB was observed at 8.05 ppm with a very lowintensity (FIG. 30), and it was found disappeared in the ¹H NMR spectrumof the 1:2 (w/w) mixture of 288K di-THY and 219K di-DAAP 1,4-PBs. Thedisappearance of the signal indicates that THY and DAAP end groups couldfind and bind with each other in CDCl₃, even when attached to chain endsof polymers of M_(w)˜200,000 g/mol. Likewise, the signal of imideprotons of the CA end groups of 200K di-CA 1.4-PB, along with those ofthe amide protons of the HR end groups of 240K di-HR 1,4-PB, were notobservable in the ¹H NMR spectrum of the 1:1 (w/w) mixture of 200K di-CAand 240K di-HR 1,4-PBs (FIG. 31). Signals of the TB end groups of 250Kdi-TB 1,4-PB were also found disappeared after the polymer was mixedwith 230K di-TA 1,4-PB in a 1:1 wt ratio (FIG. 32). These resultssuggest that all three complementary associative pairs can providesufficient strength of end-association for telechelic 1,4-PB chains ofM_(w)˜200,000 g/mol to form supramolecular aggregates stable at least onthe time scale of ¹H NMR spectroscopy.

Example 20: Shear Viscometric Study of Complementary End-Association

Shear viscometry was used as a complementary measure of ¹H NMR study toevaluate the strength of hetero-complementary pairs. 1-Chlorododecane(CDD) was chosen as the solvent due to its low interference withhydrogen bonding, low volatility at room temperature, high solvency for1,4-PB backbones, and being a pure solvent. For all of the fourhetero-complementary pairs (THY/DAAP, HR/CA, DA/DB, and TA/TB),telechelic polymers of M_(w)˜200,000 g/mol were used. In addition toCDD, dodecane and Jet-A were also used in shear viscometric study ofTHY/DAAP and HR/CA pairs, respectively. Except for di-DA and di-TA1,4-PBs, polymer solutions in 1-chlorododecane were prepared bycombining polymer and solvent at a weight fraction of polymer=1 wt % inclean 20 mL scintillation vials, which were placed on a Wrist-ActionShaker (Burrell Scientific) at room temperature for up to 16 h to allowcomplete dissolution of polymers. 1 wt % CDD solutions of di-DA anddi-TA 1,4-PBs of M_(w)˜200,000 g/mol were prepared according to theprocedure described in Examples 2-5. For each hetero-complementaryassociative pair, 1 wt % solutions of polymer mixture were prepared bymixing 1 wt % solutions of the individual polymers in desired weightratios in 20 mL scintillation vials at room temperature. Shear viscosityof polymer solutions were measured according to the procedure describedherein (see, e.g. Examples 16-17).

Steady-flow shear viscometry at 25° C. was used in parallel with ¹H NMRspectroscopy to investigate the ability of OHB-based and CAHB-basedhetero-complementary associative pairs to afford supramolecularaggregates of telechelic 1,4-PBs of M_(w)≥200,000 g/mol that are stableenough at low-moderate shear rates to provide modulation of rheologicalproperties. In other words, it is expected that at the sameconcentrations, the solution of complementary polymer pair would be moreviscous than those of individual components. To avoid possiblecomplications arising from the multi-component nature of fuels,1-chlorododecane (CDD) was chose as the model solvent, and prepared allpolymer solutions at 1 wt % in CDD. In both THY/DAAP and HR/CAcomplementary polymer pairs, none of them showed the expectedenhancement in shear viscosity due to complementary end-association(FIGS. 43 and 44). To find out if the comparatively polar CDD(dielectric constant=4.2 at 25° C.) interferes with THY/DAAP and HR/CAcomplementary interactions, the experiments were repeated in less polarsolvents: Dodecane (dielectric constant=2.0 at 20° C.) and Jet-A(dielectric constant=1.8 at 20° C.) were used for THY/DAAP pair andHR/CA pair, respectively. As shown in FIGS. 43 and 44, the expectedenhancement in shear viscosity was still absent in both cases when lesspolar solvents were used.

Different results were observed in the case of TA/TB pair. The 1:1 (w/w)mixture of 1 wt % CDD solutions of 230K di-TA and 250K di-TB 1,4-PBs wasfound considerably more viscous than both solutions (FIG. 33), and theobserved enhancement in viscosity illustrated that the strength of TA/TBcomplementary end-association was sufficient to drive the formation ofsupramolecules stable at shear rates investigated in the present study.As discussed in above, strong self-association of 230K di-TA 1,4-PBresulted in significant difference in shear viscosity between the 1 wt %CDD solution of 230K di-TA 1,4-PB and that of the non-associativepre-polymer 230K di-TE 1,4-PB (FIG. 33). It was observed that theaddition of equal amount (by weight) of 250K di-TB 1,4-PB furtherenhanced the shear viscosity. What is also worth noting is theshear-thinning behavior observed in the 1 wt % CDD solution of 1:1mixture of 230K di-TA and 250K di-TB 1,4-PBs, which is a feature sharedby aqueous solutions of water-soluble telechelic associative polymers[48-51]. As for the 1 wt % CDD solution of 250K di-TB 1,4-PB, eventhough GPC-LS analysis confirmed no crosslinking of polymer backbonetook place during end-functionalization with tertiary amine groups, itwas found that it was more viscous than that of the non-associative 230Kdi-TE 1,4-PB. Aggregation of triazole units resulting from theend-functionalization reaction (FIG. 45) may contribute to the abovedifference in shear viscosity [52].

With the positive results of the pair of 230K di-TA/250K di-TB 1,4-PBs,the viscometric study was extended further to the complementary DA/DBassociation as an attempt to approach the limit of the strength ofcarboxyl/tertiary amine association. FIG. 34 shows the results of 1 wt %CDD solutions of the corresponding polymers (230K di-DE, 230K di-DA, and250K di-DB 1,4-PBs) and the 1:1 (w/w) DA/DB mixture. Surprisingly,strong enhancement in shear viscosity induced by complementary DA/DBassociation was still observed in the 1:1 mixture. While onlyinsignificant difference in shear viscosity was observed between the 1wt % CDD solution of 230K di-DA 1,4-PB and that of the non-associative230K di-DE 1,4-PB, the considerable increase in viscosity due to DA/DBcomplementary end-association reaffirmed the promising strength ofcarboxyl/tertiary amine interaction.

The final part of the shear viscometric study of carboxyl/tertiary aminepairs was to investigate if the TA/TB complementary end-association waseffective in Jet-A when the M_(w) of the 1,4-PB backbone increased to430,000 g/mol, and the results are shown in FIG. 35. Strong enhancementin shear viscosity due to TA/TB complementary association was observed:At 1 wt %, the 1:1 mixture of 430K di-TA and 430K di-TB 1,4-PBs in Jet-Awas found significantly more viscous than the Jet-A solutions of theindividual polymers. These results indicate that when used in dendriticconfigurations, carboxyl/tertiary amine pair is suitable for buildingcomplementary pairs of telechelic associative polymers as mist-controladditives for fuels.

Example 21: A.1 Measurements of Polymer Molecular Weights

The determination of molecular weight and molecular weight distributionis of central interest in polymer analysis, as the molecular weight of apolymer directly relates to its physical properties.[53] Take telechelicassociative polymers as mist-control additives for kerosene for example,their efficacy in providing fire protection and resistance to sheardegradation rely on proper choice of backbone length, which falls in therange M_(w)=5×10⁵-10⁶ g/mol. Table 8, which shows molecular weightmeasurement methods, summarizes common characterization methods fordetermining different average molecular weights (MWs) and molecularweight distributions (MWDs) of polymers [34, 53, 54].

TABLE 8 Method Absolute Relative M_(n) M_(w) Range (g/mol) Proton NMRend-group × × M_(n) < 2.5 × 10⁴ analysis Vapor pressure × × M_(n) < 3 ×10⁴ osmometry Ebulliometry × × M_(n) < 3 × 10⁴ Light Scattering (LS) × ×10⁴ < M_(w) < 10⁷ Intrinsic Viscosity × M < 10⁶ GPC^(a) with × × × 10³ <M_(w) < 10⁷ concentration detectors GPC^(a) with × × × 10⁴ < M_(w) < 10⁷concentration and LS detectors MALDI-TOF-MS^(b) × × × M < 3 × 10⁴^(a)GPC, gel permeation chromatography. ^(b)MALDI-TOF-MS,matrix-assisted laser desorption/ionization time-of-flight massspectroscopy

Among the methods in Table 8, GPC with concentration and LS (lightscattering) detectors (referred to as “GPC-LS” herein) was chosen in thepresent study for determining MW and the MWD of telechelic associative1,4-PBs due to the following reasons: (1) it allows measurements ofabsolute weight-average MWs and corresponding MWDs; (2) it has a wideapplicable range (104-10⁷ g/mol) which covers the MW range of interest(5×10⁵-10⁶ g/mol) for mist-control applications; (3) it is comparativelyeasy to implement. Although MALDI-TOF-MS is capable of measuringabsolute MWs and MWDs of polymers with more accuracy than GPC-LS, it isnot as useful in analyzing polymers of MW>30,000 g/mol [55]; selectionof matrix compounds, sample preparation and interpretation of the massspectra become difficult in the case of synthetic polymers of MW>30,000g/mol and thus detract from the benefits associated with the unrivalledaccuracy provided by MALDI-TOF-MS [53, 54, 56]. Given that manyassociative polymers as herein described are telechelic 1,4-PBs ofMW>>30,000 g/mol, it is clear that GPC-LS can be a better option tomeasure MWs than MALDI-TOF-MS in the present study. The same rationalealso applies to the other competing method, proton NMR end-groupanalysis, which has been widely used in determining number-average MWs(i.e., M_(n)) of synthetic polymers via comparing the integration valuesof signals of backbone protons to those of the end-group protons [53,57, 58]. The implementation of proton NMR end-group analysis can bestraightforward: the M_(n) value of a polymer can be derived from its ¹HNMR spectrum without any additional experimental work. However, thedetermination of M_(n) by proton NMR end-group analysis for polymers ofMW>25,000 g/mol loses its accuracy due to a diminished resolutionresulting from the inherent detection limit of proton NMR spectroscopy,and the uncertainty in the M_(n) values becomes greater for polymers ofhigher MWs [53]. The other issue of this method is that it lacks theability to measure molecular weight distributions (MWDs) of polymers.These shortcomings render proton NMR end-group analysis a less effectivemethod to characterize high-MW (i.e., MW>100,000 g/mol) telechelic1,4-PBs as potential mist-control additives for kerosene.

In the case that associative groups are attached onto the chain ends oftelechelic 1,4-PBs, measuring of MWs and MWDs of such polymers by GPC-LSbecomes challenging, since the associative chain ends could possiblyinteract with the column packing, or drive the formation ofsupramolecular aggregates in THF, leading to false reading of MWs andMWDs. It was found that compared to the non-associative 230K di-TE1,4-PB, the apparent M_(w) of 230K di-TA 1,4-PB was found to be higherby 63% (see Table 9, which shows molecular weight and PDI(polydispersity index) data of tert-butyl ester- and carboxyl-terminatedtelechelic 1,4-PBs, and FIG. 23).

TABLE 9 N = 1 N = 2 N = 4 N = 8 N = 4 Before TFA M_(W) 226 230 230 207430 Hydrolysis (kg/mol)^(a) 1.43 1.53 1.50 1.43 1.49 PDI^(b) After TFAM_(W) 276 299 375 304 510 Hydrolysis (kg/mol)^(a) 1.56 1.73 1.72 1.511.61 PDI Increase in 22.12 30.00 63.04 46.86 18.60 M_(W) (%)^(a,b)determined by GPC-LS

It was hypothesized that the apparent increase in M_(w) resulted fromthe aggregate of associative TA end groups in THF, rather thancrosslinking of 1,4-PB backbone during TFA hydrolysis of tert-butylester groups. To test the hypothesis, 230K di-TA 1,4-PB was treated withLiAlH₄ in THE so as to reduce the highly associative carboxyl groups toless associative hydroxyl groups. The GPC-LS result of the resultanthydroxyl-terminated 230K telechelic 1,4-PB, as shown in FIG. 36,virtually overlaps with that of 230K di-TE 1,4-PB, although the formerseems slightly broadened compared to the latter. Comparison of the threeGPC-LS traces in FIG. 36 verified the hypothesis: the apparent increasein M_(w) after TFA hydrolysis of 230K di-TE 1,4-PB was due toaggregation of associative TA end groups, since the increase in M_(w)disappeared after the carboxyl groups on polymer chain ends were reducedto hydroxyl groups. It also suggests that the mild condition of TFAhydrolysis does not cause appreciable amount of crosslinking of 1,4-PBbackbone. As for the broadening of GPC-LS trace of hydroxyl-terminated230K telechelic 1,4-PB, it is thought to result from interaction ofhydroxyl-terminated chain ends with column packing. The results in FIG.36 also reveal the importance of interpreting GPC-LS results oftelechelic associative polymers with scrutiny, since association ofchain ends and chain-end/column interaction can both result in falsereading of MWs and MWDs. In other words, using the non-associative formsof telechelic associative polymers in GPC-LS analysis yields moreaccurate information concerning the MWs and MWDs of polymer backbones onthe condition that the transformation of associative chain ends tonon-associative counterparts does not damage the backbones.

Example 22: Effect of COD Purity on the Proceeding of ROMP with CTAs

It was found that the purity of VCH (4-vinylcyclohexene)-free COD has aprofound effect on the synthesis of telechelic 1,4-1,4-PBs via ROMP ofCOD using Grubbs II: peroxides and n-butanol (introduced during BH₃.THFtreatment of COD according to the Macosko protocol) can also adverselyaffect the metathetical activity of Grubbs II by reacting with it andirreversibly transforming it into inactive species. In response to theissues associated with peroxides and n-butanol, a multi-stage process(Section 2.2.3) was developed to rigorously purify COD.

In particular, in an exemplary purification procedure, redistilled cis,cis-1,5-cyclooctadiene (COD, 72.3 g, 0.67 mol) was syringe-transferredto a 250 ml Schlenk flask in an ice bath at 0° C. under argonatmosphere. Under argon flow, 1M borane-THF complex in THE (BH₃.THF, 108mL, 0.11 mol) was then slowly added into the flask over a 10-min period.The flask was taken out of the ice bath, and left to stir under argonatmosphere at room temperature for 2 h. THE was evaporated under reducedpressure at room temperature to an extent that the concentration ofresidual THE in the mixture was below 300 ppm (verified by ¹H NMRanalysis). The monomer was vacuum distilled from the mixture at 40° C.,100 mTorr into a 100 mL Schlenk flask (loaded with 9 g of MAGNESOL® xland a magnetic stir bar) in a dry-ice tub. The mixture was stirred underargon atmosphere at room temperature overnight. The monomer was vacuumdistilled again at 45° C. and 100 mTorr from the mixture into a 100 mLSchlenk flask (loaded with 10 g of calcium hydride (CaH₂) and a stirbar) in a dry-ice tub in order to remove moisture introduced byMAGNESOL® xl. After stirring at room temperature for 3 h under argonflow, the monomer was once again vacuum distilled (45° C., 100 mTorr)from the mixture into a 100 mL Schlenk flask in a dry-ice tub. Afterwarmed to ambient temperature, the flask was sealed with a SUBA-SEAL®rubber septum while argon stream was flowing, and placed in a freezer at−30° C. for storage of purified COD (40.0 g, 55.3% yield). The purifiedmonomer was vacuum-distilled again at 35° C. prior to use.

To illustrate the influence of the purity of VCH-free COD on thepreparation of telechelic 1,4-PBs via ROMP of COD, the synthesis ofdi-TE 1,4-PB via the two-stage ROMP of COD with octa-functionaltert-butyl ester-terminated bis-dendritic CTA (compound 8 in FIG. 46B)was chosen as the benchmark reaction (FIG. 37). Two different batches ofVCH-free COD were prepared: the first (i.e., the control, COD I) wasafforded via purification according to only the Macosko protocol,whereas the second one (COD II) was prepared according to thepurification procedure described above. The implementation of two-stageROMP using both batches of COD was the same as the purificationprocedure described above, in which the total monomer:CTA ratio was2000:1, and 100 eq of COD was used in the first stage of ROMP; the loadof Grubbs II was 1/30 eq of the CTA. Here the following properties toquantitate the effect of the purity of COD were chosen: (1) the periodof time during which the reaction mixture develops enough viscosity tostop the magnetic stir bar from moving after the addition of 1900 eq ofCOD (t_(v)) (2) the overall conversion of COD (X_(f), measured by ¹H NMRanalysis of the aliquot of reaction mixture) (3) the cis/trans ratio ofthe polymeric species in the aliquot (measure by ¹H NMR analysis) (4)M_(w) of the resultant polymer (measured by GPC-LS). The results for CODI and COD II were summarized in Table 10, which shows the results ofsynthesis of di-TE 1,4-PB via ROMP of batch 1 and batch 2 VCH-free COD.

TABLE 10 COD I COD II t_(v)(min) 40.0  1.5 X_(f) (mol %) 85.0 97.6cis/trans ratio   2.20   1.73 M_(w) (kg/mol) 264    142    PDI   1.58  1.43

Table 10 shows that the second stage of ROMP of COD II proceededsignificantly faster (t, =1.5 min) compared to that of COD I (t, =40min); the conversion of COD II was nearly quantitative (X_(f)=97.6%),whereas the reaction stopped at X_(f)=85% in the case of COD I. Inaddition, ¹H NMR analysis of aliquots taken in the end of polymerizationreactions also revealed that the use of COD II led to a lower cis/transratio (1.73) compared to the case of COD I (2.20). The M_(w) of theresultant polymer of ROMP of COD II (142,000 g/mol), as revealed byGPC-LS analysis, was found significantly lower than that of ROMP of CODI (264,000 g/mol). When considered as a whole, these results indicatethat Grubbs II possesses a higher metathetical activity (or a higherturnover number) when impurities in VCH-free COD that can interfere withGrubbs II are removed. This explains the much faster reaction rate ofthe second stage of ROMP of COD II. Similarly, Grubbs II in the presenceof COD II can perform more cycles of metathesis reactions compared to inCOD I, and thus a nearly quantitative X_(f)=97.6% was achieved in thecase of COD II. The low cis/trans ratio (1.73) and M_(w) (142,000 g/mol)resulting from ROMP of COD II suggest that a considerable fraction ofruthenium complexes on polymer chain ends remained metathetically activewhen COD II was mostly consumed, and as a result they continued to reactwith available C═C bonds present in the reaction mixture (in this case,C═C on polymer backbones) till they reached their maximum turnovernumber. The consumption of backbone by active ruthenium centers on chainends (i.e., back-biting) led to the decreases in cis/trans ratio and MW.

In sum, the enhanced activity of Grubbs II observed above validates themulti-stage purification procedure of COD described above.

Example 23: Example of Controlling Drag Reduction

In some embodiments, the associative polymers described herein can beused to provide a composition in which the property controlled is dragreduction. In particular, using the methods described herein, thecomposition can have a more than 10% reduction in the pressure droprequired to drive a given volumetric flow rate through a given pipeline.

In particular, a skilled person can identify the non-polar host to betransported in which the drag is desired to be reduced.

The skilled person can then use published solubility parameters toestimate the solubility parameter of the identified non-polar host, orin the alternative, the skilled person can use literature on polymersolubility in similar liquids, and use this information to identifypolymers that would be expected to dissolve in the non-polar host, foruse as backbones of the associative polymers. The solubility can beconfirmed by the skilled person by using techniques identifiable to theskilled person, for example by dissolving a sample of the polymer in thehost and determining if it is homogeneous (e.g., by performinglight-scattering measurements).

The skilled person can then use published dielectric constants toestimate the dielectric constant of the host liquid, and determine thekind of associative interaction of the FGs would be most suitable. Forexample, if the dielectric constant is less than or approximately 2,there are a wide range of suitable associative groups, includingordinary hydrogen bonding moieties (e.g. Hamilton receptor/cyanuric acidpairs, thymine/diacetamidopyridine pairs, and other identifiable to askilled person) and charge transfer complexing moieties (e.g.dinitrophenyl/carbazole pairs and other identifiable to a skilledperson). As the dielectric constant increases, the range of viableassociative moieties decreases. For example, in chlorododecane(dielectric constant of 4.2 at 25° C.), charge-assisted hydrogen bondingmoieties perform better than ordinary hydrogen-bond moieties. If thereare organic acids (such as, Butyric acid, isobutyric acid, valeric acid,isovaleric acid, Heptanoic acid, and others identifiable to a skilledperson) or organic bases (trimethylamine, diethylamine,diisopropylamine, Triethylamine, Diisobutylamine, diisoamylamine,diphenylamine, and others identifiable to a skilled person) present inthe host composition, ionic interactions or ligand-metal interactions (atype of Brønsted/Lewis acid/base interaction) can be more suitable thancharge-assisted hydrogen bond association. Therefore, some additionaloptimization can be performed as described below.

The additional optimization can be performed by preparing severaltelechelic polymers with backbone degree of polymerization of at least200 and with candidate associative groups at their ends (e.g. ordinaryhydrogen bonding moieties and/or charge transfer complexing moieties),and dissolving them in the host liquid using polymer concentrationapproximately equal to the overlap concentration for the backbonepolymer and length used in the trial polymers (e.g., by calculating c*as described herein). The polymers that do not dissolve can beidentified, and their corresponding associative end groups can bedesignated as being unsuitable, to thereby identify the suitableassociative groups. If the viscosity of the non-polar composition is notgreater than it would be for a solution of a non-associative polymer ofthe same backbone, length and concentration, the associative end groupscan be modified by increasing the number of associative moieties in eachgroup (i.e., increase the strength of association using polyvalentinteractions).

Using one or more of the combinations of polymer backbone structure andend-group structure identified above, the skilled person can thenestimate the backbone length that is compatible with a desirable oracceptable polymer concentration in the host. For example, if thebackbone is determined to be polybutadiene, and the associative polymerconcentration needs to be kept down to 0.8% or less (the “x” marked onthe vertical axis of FIG. 40), then the minimum polybutadiene backbonecan be read off a graph of the relationship between the overlapconcentration and the weight-average molecular weight (as shown by thehorizontal line from the “x” on the vertical axis to the correspondingpoint on the c* vs M_(w) relationship for polybutadiene and the verticalline from that point down to the horizontal axis in FIG. 40), leading toa value of M_(w) of about 400,000 g/mol.

A skilled person can then use experiments to refine the choice ofbackbone, backbone length, and FGs by preparing candidate polymers withthe most promising backbone, backbone length, and FGs, then subjectingthem to a limited set of experiments to validate their performance inboth reducing turbulent drag (e.g., measuring the flow rate of thenon-polar composition though a conduit, or measuring the change inpressure of the non-polar composition flowing through a conduit) and, ifdesired, resisting degradation due to turbulent flow (e.g. by measuringchanges in viscosity of the non-polar composition after transportationthrough a conduit). If the required concentration is found by theskilled person to be too high (e.g. the amount of polymer required wouldbe too costly), then the skilled person can prepare another polymer withthe same, but longer, backbone and repeat the process until the polymershows efficacy at an acceptably low concentration. This exemplaryprocedure is expected to give a drag reduction in turbulent pipe flow ofat least 10%. If the extent of drag reduction is less than 30%, theskilled person can improve drag reduction up to 30% by increasing thestrength of association, for example by increasing the number ofassociative moieties per associative group (e.g., using end groups withfour carboxyl groups rather than two) or by using a stronger type ofassociation (e.g., using charge-assisted hydrogen bonding—that is, ahydrogen bond formed between a hydrogen bond donor and hydrogen bondacceptor where the hydrogen bond donor is more acidic than the conjugateacid of the hydrogen bond acceptor by at least about 4 pKa units-ratherthan ordinary hydrogen bonding—that is, a hydrogen bond formed between ahydrogen bond donor and hydrogen bond acceptor where the hydrogen bonddonor is less acidic than the conjugate acid of the hydrogen bondacceptor).

Example 24: Use of Associative Polymers in a Fuel in an Engine whileMaintaining Engine Performance

In this example, an exemplary self-associative polymers wereincorporated in fuel at a level that is appropriate for drag reductionand/or mist control for improved fire safety. 430K di-TA PB was selectedas the test polymer along with diesel as the base fuel; a polymerconcentration of 0.1 wt % in diesel was subsequently chosen. Aconcentrated 1 wt % stock solution of the exemplary associative polymerwas prepared by mixing the polymer with diesel under oxygen-freecondition at 120° C. for 12 hours, and two identical 0.1 wt % dieselsolutions of the polymer with a volume of 1.3 liters were prepared bydiluting the 1 wt % stock solution with the same base fuel at roomtemperature. Test samples comprised the two 0.1 wt % solutions and two1.3-liter bottles of unmodified base fuel as controls. A 3.75 kW dieselgenerator connected to a Simplex Swift-e load bank and a Fluke 434Series II Energy Analyzer was used as the test apparatus, and the testswere performed at the Vehicle Emission Research Laboratory (VERL) of theCenter for Environmental Research & Technology (CE-CERT), University ofCalifornia at Riverside. A sequence of generator load/operating timecomprising the following stages was used to carry out the tests: 2000Watts (˜53% of its rated power)/9 min, 3000 Watts (˜80% of the ratedpower)/9 min, 3500 Watts (˜93% of the rated power)/6 min, 3000 Watts/9min, and 2000 Watts/9 min. Between samples the fuel supply to the enginewas switched to a reservoir filled with the reference fuel (the samediesel fuel that was used to prepare the samples with associativepolymers herein described) to keep the generator operating. The ACoutput from the generator was recorded continuously by the EnergyAnalyzer, and the emissions were analyzed using gas analysis of anisothermal stream of precisely calibrated dilution of the exhaust gas;quantitative values for carbon dioxide (CO₂), carbon monoxide (CO),mono-nitrogen oxide (NOx), methane (CH₄) and total hydrocarbons (THC)were continuously monitored. Samples were run in a blind randomizedsequence and the results were quantitatively analyzed prior to unmaskingthe sample identification. The results show no decrease in power outputat any of the three loads to within the uncertainty of the powermeasurement. The results showed no adverse effects on engine emissions(Table 11). For the composition used in this example, it was notpossible to identify the time at which the fuel supply to the engine wasswitched between the reference fuel, since none of the measuredquantities changed at or near the time the valve was switched. Theemissions of CO and THC were reduced (11), while the power output wasthe same (to within the uncertainty of the measurement) as for untreateddiesel.

TABLE 11 % change Condition A #29a CO₂ Sample-Diesel 2 kW 2.03Sample-Diesel 3 kW −0.09  Sample-Diesel 3.5 kW 0.43 Sample-Diesel 3 kW1.56 Sample-Diesel 2 kW 1.46 CO Sample-Diesel 2 kW 5.63 Sample-Diesel 3kW −4.34  Sample-Diesel 3.5 kW −10.20   Sample-Diesel 3 kW −1.93 Sample-Diesel 2 kW 8.87 THC Sample-Diesel 2 kW −15.54   Sample-Diesel 3kW −13.04   Sample-Diesel 3.5 kW −11.54   Sample-Diesel 3 kW −8.73 Sample-Diesel 2 kW −0.68  NO_(×) Sample-Diesel 2 kW 4.30 Sample-Diesel 3kW 2.81 Sample-Diesel 3.5 kW 3.76 Sample-Diesel 3 kW 4.13 Sample-Diesel2 kW 5.96 ^(a)A#29 is diesel treated with 0.1 wt % di-TA PB

Example 25: Reduction of Emissions in Fuels Comprising AssociativePolymers

In this example, exemplary donor-acceptor polymers are incorporated infuel at a level that is appropriate for drag reduction and/or mistcontrol for improved fire safety, with the additional benefit thatemissions from the engine are reduced. A 1:1 (w/w) mixture of 630K di-DAPB and 540K di-DB PB was selected as an exemplary donor-acceptor polymerpair along with diesel as the base fuel; a total polymer concentrationof 0. wt % in diesel was subsequently chosen. A concentrated 1 wt %stock solution of the donor-acceptor pair was prepared by mixing thepair with diesel at room temperature for 12 hours and at 70° C. for 7hours, and two identical 0.1 wt % diesel solutions of the pair with avolume of 1.3 liters were prepared by diluting the 1 wt % stock solutionwith the same base fuel at room temperature. Test samples comprised thetwo 0.1 wt % solutions and two 1.3-liter bottles of unmodified base fuelas controls. The Same apparatuses, procedures, and characterizationsdescribed in Example 24 were used in this example. Samples were run in ablind randomized sequence and the results were quantitatively analyzedprior to unmasking the sample identification. The results showed nodecrease in power output at any of the three loads to within theuncertainty of the power measurement. For the composition used in thisexample, the emissions of CO and THC were reduced (Table 12), while thepower output was the same (to within the uncertainty of the measurement)as for untreated diesel.

TABLE 12 % change Condition AB #90^(a) AB #8^(a) AB average CO₂Sample-Diesel 2 kW  0.68  0.95  0.81 Sample-Diesel 3 kW −1.74  1.40−0.17 Sample-Diesel 3.5 kW  0.71  0.92  0.82 Sample-Diesel 3 kW  0.19−0.43 −0.12 Sample-Diesel 2 kW  0.09  1.09  0.59 CO Sample-Diesel 2 kW−13.89  −10.99  −12.44  Sample-Diesel 3 kW −15.81  −12.52  −14.16 Sample-Diesel 3.5 kW −14.36  −16.31  −15.33  Sample-Diesel 3 kW −10.79 −14.91  −12.85  Sample-Diesel 2 kW −11.79  −12.49  −12.14  THCSample-Diesel 2 kW −25.12  −23.83  −24.47  Sample-Diesel 3 kW −14.39 −16.65  −15.52  Sample-Diesel 3.5 kW −10.13  −12.63  −11.38 Sample-Diesel 3 kW −11.75  −12.50  −12.12  Sample-Diesel 2 kW −12.27 −13.37  −12.82  NO_(×) Sample-Diesel 2 kW −1.29  0.77 −0.26Sample-Diesel 3 kW −3.16 −0.35 −1.76 Sample-Diesel 3.5 kW −2.17 −0.59−1.38 Sample-Diesel 3 kW −1.95 −0.43 −1.19 Sample-Diesel 2 kW  0.77 2.70  1.73 ^(a)AB #90 is a first sample of 0.1 wt % 1:1 di-DA PB/di-DBPB; AB #90 is a second sample of 0.1 wt % 1:1 di-DA PB/di-DB PB

Based on the observed reductions of THC and CO, a corresponding increasein fuel efficiency occurred.

Example 26: Improvement of Fuel Efficiency with Self-AssociativePolymers

The emissions data discussed for Example 24 (0. wt % diesel solution of430K di-TA PB) show a reduction in THC and CO emissions compared to thediesel reference sample, indicating a more efficient burning of thefuel.

Example 27: Improvement of Fuel Efficiency with Donor-AcceptorAssociative Polymers

The emissions data discussed for example 25 (0.1 wt % diesel solution of630K di-DA PB/540K di-DB PB 1:1 mixture) show a reduction in THC and COemissions compared to the diesel reference sample, indicating a moreefficient burning of the fuel.

Example 28: Additional Improvement of Fuel Efficiency withDonor-Acceptor Associative Polymers

The exhaust gas temperatures for untreated diesel and the sampledescribed in Example 25 (0.1 wt % diesel solution of 630K di-DA PB/540Kdi-DB PB 1:1 mixture) were measured by a thermal couple immediatelyafter the exhaust was diluted with an isothermal stream of carrier gas(hence, the temperature of the actual exhaust gas was considerablyhigher that reported here after dilution). The results revealed a 5° C.reduction for the exhaust corresponding to example 25, indicating a moreefficient burning and conversion of fuel energy to useful power in theengine for this example.

Example 29: Materials

All chemical reagents were obtained at 99% purity from Sigma-Aldrich,Alfa Aesar, or Mallinckrodt Chemicals. Magnesol® XL was purchased fromThe Dallas Group of America, Inc. ¹H-NMR spectra were obtained using aVarian Inova 500 spectrometer (500 MHz); all spectra were recorded inCDCl₃. Chemical shifts were reported in parts per million (ppm) and werereferenced to residual protio-solvent resonances. Deuterated solventsused for ¹H-NMR and SANS experiments (CDCl₃ and d12-cyclohexane) werepurchased from Cambridge Isotope Laboratories. Cylindrical quartz“banjo” cells used in scattering experiments were purchased from HellmaAnalytics.

Example 30: Representative Procedure for Purification of Cyclooctadiene(COD)

Trace impurities introduced when purifying COD to remove itsconstitutional isomer 4-vinylcyclohexene using borane-tetrahydrofurancomplex (BH₃.THF) were found to affect the procedure to prepare longtelechelic polycyclooctadienes (PCODs).

Redistilled-grade COD (72.3 g, 0.67 mol) was syringe-transferred to a250 ml Schlenk flask in an ice bath under argon. 1 M BH₃.THF complex inTHE (108 ml, 0.11 mol) was slowly added into the flask over 10 min. Theflask was taken out of the ice bath, and left to stir under argon atroom temperature for 2 h. THE was evaporated under reduced pressure atroom temperature to an extent that the concentration of residual THF inthe mixture was below 300 ppm (verified by ¹H NMR analysis). The monomerwas vacuum distilled from the mixture at 40° C. into a 100 ml Schlenkflask (loaded with 9 g of Magnesol® XL and a stir bar) in a dry-ice tub.The mixture was stirred under argon atmosphere at room temperatureovernight. The monomer was vacuum distilled from the mixture into a 100ml Schlenk flask (loaded with 10 g of calcium hydride (CaH₂) and a stirbar) in a dry-ice tub. After stirring at room temperature for 3 h underargon flow, the monomer was vacuum distilled from the mixture into a 100ml Schlenk flask in a dry-ice tub. After being warmed to ambienttemperature, the flask was sealed with a Suba-Seal rubber septum whileargon was flowing through the flask, and placed in a freezer at −30° C.for storage of the rigorously purified COD (40.0 g, 55.3% yield). Therigorously purified monomer was vacuum distilled again prior to use.

FIG. 60 shows ¹H NMR spectra of increasingly purified COD in the rangefrom 3.4 to 5.9 ppm. FIG. 60, Panel A, COD after BH₃.THF treatment andvacuum distillation (containing ˜330 ppm of butanol based onintegration). FIG. 60, Panel B, Alternatively, COD further purified withmagnesium silicate/CaH₂ treatments (to show removal of butanol and theresulting purity of COD used as monomer).

Example 31: GPC-MALLS for Characterization of Polymers

MALLS, i.e. Multi-angle Laser Light Scattering, was used in conjunctionwith GPC to determine the molecular weights and polydispersity of thepolymers. The system used a Wyatt DAWN EOS multi-angle laser lightscattering detector (λ=690 nm) with a Waters 410 differentialrefractometer (RI) (λ=930 nm) connected in series. Chromatographicseparation by the size exclusion principle (largest comes out first) wasachieved by using four Agilent PLgel columns (pore sizes 10³, 10⁴, 10⁵,and 10⁶ Å) connected in series. Degassed THF was used as the mobilephase with a temperature of 35° C. and a flow rate of 0.9 ml/min. Thetime for complete elution through the system was 50 min, and MALLS andRI data were recorded at 5 Hz.

Samples were prepared by dissolving 5 mg of polymer in 1 ml of THE andfiltering the solution through 0.45 m PTFE membrane syringe filtersimmediately before injection. An injection volume of 20 μl was used. Thedata were analyzed by Wyatt Astra Software (version 5.3.4) using theZimm fitting formula with dn/dc=0.125 for PCOD in THF to obtainweight-average molecular weight (M_(w)) for each polymer reported.Polymers are described in the Table 13.

TABLE 13 Characterization of polymers in this application. M_(w) ^(a)M_(n) ^(a) M_(w) ^(b) Polymer (kg/mol) (kg/mol) PDI^(a) (kg/mol) 45kNA 48.5  31.3 1.55 45kDA  44.7  28.6 1.56 45kDB  48.8  36.7 1.33 140kNA138.5  89.8 1.54 140kDA 143.1  90.2 1.59 140kDB 148.0 100.0 1.48 300kNA318.4 213.5 1.49 300kDA 304.3 201.3 1.51 300kDB 290.1 198.3 1.46 320 ±20 670kNA 637.5 441.0 1.45 670kDA 671.4 445.5 1.51 670kDB 629.2 436.21.44 600 ± 50 76kNA  76.2  52.3 1.46 76kTA  91.2  57.0 1.60 230kNA 232.8155.4 1.50 230kTA 374.5 218.7 1.71 430kNA 430.0 288.6 1.49 430kTA 510.0316.8 1.61 (^(a): determined by GPC-MALLS in THF;) (^(b): determined bybatch-mode MALLS in cyclohexane.)

Example 32: Rheology

Polymers were dissolved by shaking with tetralin, cyclohexane or Jet-A.To confirm that the end-association among telechelics is responsible forthe changes in fluid properties, additional controls were prepared bytreating some associative telechelic solutions (1.76 mg/ml) with 2.5μl/ml triethylamine (TEA) to block their end association. Shear-flowrheology data were obtained at 25° C. with stress-controlled rheometerTA AR1000, equipped with a cone-plate geometry (angle 1°, diameter 60mm) for polymer solutions in tetralin and Jet-A, and a strain-controlledrheometer TA ARES-RFS, equipped with a cone-plate geometry (angle 2°,diameter 50 mm) and a solvent trap for polymer solutions in cyclohexane,with shear rate ranging from 1000 s⁻¹ to 10 s⁻¹. For polymer solutionsin tetralin, the viscosities measured by AR1000 and ARES-RFS werechecked to be agreed with each other well. The specific viscosity valuesshown in FIG. 48 were averaged over data points taken from the rangethat doesn't have shear rate dependence (e.g., the range is 300 to 10s⁻¹ in FIG. 49A ‘DA/DB’). Three replicates with freshly preparedsolutions were measured to obtain the error bars (SD values).

Example 33: Stead-Flow Shear Viscometry

Polymers were dissolved by shaking with solvents of interest (tetralinand Jet-A). Steady shear viscosity was measured in a cone-plate geometry(60 mm diameter aluminum, 1° cone, 29 m truncation) at 25° C. using anAR1000 rheometer from TA Instruments (temperature controlled at 25° C.).Test solutions were probed in the shear rate range 1-3000 s⁻¹logarithmically (5 shear rates per decade). All viscosity data werereported in terms of specific viscosity(η_(sp)≡(η_(solution)−η_(solvent))/η_(solvent), where η_(solvent)=2.02mPa·s for tetralin and 1.50 mPa·s for Jet-A at 25° C.) which reflectsthe contribution of the polymer to the viscosity.[34]

Example 34: Shear Stability Test

A recirculation setup consisting of a Bosch 69100 In-line Electric FuelPump and a MW 122A 2AMP Regulated DC Power Supply (LKD Ind.) at 12 V(shown in FIG. 66 Panel A) was used to subject polymer solutions to aflow history that mimics, for example, recirculation of fuel through anengine's heat transfer system. Test samples were recirculated throughthe setup at room temperature for 60 s (approximately 60 passes throughthe pump using 50-60 mL of solution and a flow rate of 3 L/min). Afterrecirculation, samples were collected in 100 mL glass jars and stored at−30° C. for further tests. Between tests, the pump was rinsed 4 timeswith approximately 200 mL of hexanes, followed by drying in vacuo at 40°C. overnight to prevent cross-contamination among samples or dilution byhexanes. Shear stability was evaluated by comparing shear viscosities ofrecirculated samples to those of the corresponding unsheared controls.

Example 35: Small-Angle Neutron Scattering (SANS)

d₁₂-Cyclohexane solutions of polymers were prepared by weighing outpolymer on a Mettler precision balance (±0.01 mg) into new glassscintillation vials with PTFE lined caps and subsequently adding theappropriate amount of solvent using a precision syringe (±1%). Thesewere subsequently placed on a wrist action shaker at room temperatureovernight.

SANS data in the present application were obtained at the NationalInstitute of Standards and Technology (NIST) on beamline NG-3,preliminary experiments (data not shown) were conducted at Oak RidgeNational Laboratory (ORNL) on beamline CG-2 at the High Flux IsotopeReactor (HFIR). Samples were placed in Hellma quartz cylindrical cellswith 5 mm path length. Temperature was controlled by a recirculatingwater bath at NIST and by Peltier at ORNL. All scattering experimentswere conducted at 25° C. Two-dimensional scattering patterns were takenfor each sample using three detector distances (1.3-13 m at NIST and0.3-18.5 m at ORNL). The overall scattering vector ranges were0.003<q(Å⁻¹)<0.4 at NIST and 0.002<q(Å⁻¹)<0.8 at ORNL with the effectivelimits for a given sample determined by the signal to noise ratio.

Example 36: Multi-Angle Laser Light Scattering MALLS (“Batch Mode”)

MALLS (not connected to GPC) was used to characterize the supramolecularassembly behavior of complementary associative telechelic polymers(DA/DB mixtures) in cyclohexane. Cyclohexane solutions of polymers wereprepared by weighing out polymer on a Mettler precision balance (±0.01mg) into new glass scintillation vials (20 ml) with metal foil linedcaps and subsequently adding the appropriate amount of solvent using aprecision syringe (±1%). These were subsequently placed on a wristaction shaker at room temperature overnight. All solutions were filteredthrough 0.45 m PTFE filters into clean glass scintillation vials (20 ml)and allowed to equilibrate for at least 24 hours prior tocharacterization. MALLS measurements were carried out using a Wyatt DAWNEOS laser light scattering instrument in “batch mode” with 18 detectorsin the angular range from 22.5 to 147° using a solid-state laser (λ=690nm).

Data were acquired at 35° C. three times (rotating the vial to averageout heterogeneities) for at least 2 minutes and analyzed using WyattAstra Software (version 5.3.4). The associative supramolecules conformedto the Zimm fitting formula, which was used to evaluate the apparentweight-average molecular weight (app M_(w)) and apparent radius ofgyration (app R_(g)) for each polymer composition at each concentration,with dn/dc=0.11 for PCOD in cyclohexane.

Example 37: Modeling: a Theoretical Model of Ring-Chain Equilibrium

Statistical mechanics were used to design polymers that defyconventional wisdom by self-assembling “mega-supramolecules” (≥5,000kg/mol) at low concentration (<0.3% wt). Theoretical treatment of thedistribution of individual subunits-end-functional polymers-among cyclicand linear supramolecules (ring-chain equilibrium) predicts thatmega-supramolecules can form at low total polymer concentration-if, andonly if, the backbones are long (>400 kg/mol) and end-associationstrength is optimal (16-18 kT wherein k as used herein indicatesBoltzmann constant). Viscometry and scattering measurements of longtelechelic polymers (LTPs, M_(w)≥400 kg/mol) having polycyclooctadienebackbones and acid or amine end groups verify formation ofmega-supramolecules. They control misting and reduce drag likeultra-long covalent polymers. With individual building blocks shortenough to avoid hydrodynamic chain scission (400<M_(w) [kg/mol]≤1,000)and reversible linkages that protect covalent bonds, thesemega-supramolecules overcome the obstacles of shear degradation andengine incompatibility.

Ultra-long polymers (M_(w)≥5,000 kg/mol) exhibit dramatic effects onfluid dynamics even at low concentration (e.g., <100 ppm confers mistcontrol ([59], [7]) and drag reduction ([60]). The key to both mistcontrol and drag reduction is the ability of polymers to store energy asthey stretch, such that the fluid as a whole resists elongation. Thehigh potency of ultra-long linear polymers is due to the onset of chainstretching at low elongation rates and their high ultimateconformational elongation ([61]). For example, increasing M_(w) from 50kg/mol to 5,000 kg/mol (below, kg/mol values refer to weight-averagemolecular weight, M_(w)) decreases the critical elongation rate by morethan three orders of magnitude, and increases the ultimate molecularelongation by two orders of magnitude.

Here a set of parameter values is identified for which the equilibriumdistribution of the supramolecular species is suitable for mist-controlapplications. Based on prior literature on ultra-long polymers (whichthemselves are not acceptable due to shear degradation during routinehandling of fuel and incompatibility with engine systems),polyisobutylene chains having weight-average molecular weight ˜5×10⁶g/mol are satisfactory mist-suppressing agents at concentrations as lowas 50 ppm in kerosene [7]. Therefore, a theoretical model of ring-chainequilibrium is used to identify choices of the molecular weight oftelechelic chains (MW_(p)), the strength of end-association (εkT) andconcentration (ϕ_(total)) that would provide 50 ppm of“mega-supramolecules” (linear supramolecules of M_(w)≥5×10⁶ g/mol andcycles of M_(w)≥10×10⁶ g/mol). Initial results provide motivations tosynthesize exceptionally long telechelics and guided the selection ofassociative end groups: the model indicates that chains of approximately5×10⁵ g/mol to 1×10⁶ g/mol with ends that associate with strength 16kT-18 kT at approximately 800-1400 ppm concentration could provide thenecessary concentration of mega-supramolecules. Details of theparticular theoretical formulation were developed and described herein.

Example 38: Parameter Space for Ring-Chain Equilibrium of Long,End-Associative Telechelics

Results are presented for complementary pairs of telechelic polymers(A-A, B-B) that have similar backbone lengths (MW_(A)=MW_(B)=MW_(p)) instoichiometric solutions (ϕ_(Atotal)=ϕ_(Btotal)=ϕ_(total)/2). When the Aand B end-groups meet, they form a physical association with energy E.In the resulting parameter space of {MW_(p), ε, ϕ_(total)}, theequilibrium distribution of supramolecules are optimized formist-control applications, within the constraints on MV_(P) (M_(w)≤1×10⁶g/mol) and ϕ_(total) (<5,000 ppm) in the context of fuel additives.

The challenge associated with using end-to-end association at the lowconcentrations relevant to fuel is the tendency to form small cyclicspecies that, in effect, consume most of the telechelic building blockswithout contributing to mist control. To reduce the fraction oftelechelic chains incorporated into cyclic species, very longtelechelics are used, which reduce the fraction of polymer “wasted” insmall rings because the loop closure probability scales as N^(−3/2) forGaussian chains and N^(−1.66) for swollen chains (see Example 45). Basedon prior literature on shear degradation, MW_(p)=1×10⁶ g/mol isconsidered as an upper bound and it is compared to chains that are halfthat length MW_(p)=0.5×10⁶ g/mol to quantify sensitivity to MW_(p). As afurther step to mitigate formation of small rings, complementaryassociation of A-A and B-B telechelics is used, for which the smallestring is a dimer. This reduces the amount of telechelic wasted in smallcyclics due to two effects: i) the entropy penalty for ring closure forthe smallest possible ring is much greater than the penalty for closinga ring of size a single telechelic because the loop is twice as long asa single telechelic; and ii) the odd rings (cycles of 1, 3, 5, . . .telechelics) are eliminated, greatly reducing the fraction of buildingblocks partitioned in cyclic species.

Another challenge in the context of fuel additives is the need tomaintain low viscosity. The longer a polymer is, the lower theconcentration ϕ* at which the chains begin to overlap and viscositybegins to increase strongly. Therefore, the total polymer concentrationϕ_(total) needs to be less than the overlap concentration of theindividual telechelic building blocks, ϕ*(MW_(p)). Further, the modelshows that a concentration of ϕ_(total)=¼ϕ*(MW_(p)) is low enough thatall supramolecules in the equilibrium distribution are below theirrespective overlap concentrations. For the two chain lengths selectedabove, results are presented for ϕ_(total)=¼ϕ*(MW_(p)), which is 800 ppmfor MW_(p)=1×10⁶ g/mol, and 1400 ppm for MW_(p)=0.5×10⁶ g/mol. Inaddition, results are presented for 1×10⁶ g/mol at 1400 ppm, both toillustrate the change in the distribution of supramolecular species withincreasing concentration (from ϕ_(total)=800 ppm to 1400 ppm forMW_(p)=1×10⁶ g/mol) and to illustrate the effect of the size of thetelechelic building blocks at matched total concentration (comparingboth chain lengths at ϕ_(total)=1400 ppm). With the above choices forMW_(p) and ϕ_(total), the problem is reduced to a single dimension, c,which is examined over its physically relevant range: it is much greaterthan kT to drive association and it is much less than 150 kT(approximate strength of a covalent bond) so that the reversible linkscan function as tension relief links that protect against chain scissionin strong flows.

TABLE 14 Literature Values for Parameters in the Model Polydienebackbone double non-1,4- ρ b M_K 1,4-rich cis trans units (b) [g/ml] C∞[nm] [g/mol] References Polyisoprene 100% (a) 0 (a) 0 (a) 0.910 5.2 0.93128.6 [1]  70% 23%  7% 0.900 4.7 0.84 119.6 [2]  19% 45% 36% 0.895 5.30.96 156.9 [1], [4] Polybutadiene  98%  1%  1% 0.900 4.6 0.83 90.5 [1],[3]  29% 53% 18% 0.895 5.6 1.01 119.8 [1], [5]  44% 59%  7% 0.895 5.50.99 112.5 [1], [5] Properties measured at temperature of 25° C. ρdensity in the melt state C∞ characteristic ratio b Kuhn step length M_Kmolar mass of a Kuhn segment Cells in the right top row forpolybutadiene starting from the cell of ′98%′ through end of the row:most similar to the polycyclooctadiene in this study, which has 75% cisunits, 25% trans units and no 1,4-units. Notes: (a) natural rubber (b)for polyisoprene, these are mainly 3,4-units and trace amounts of1,2-units; and for polybutadiene, these are 1,2-units

Example 39: Modeling: Computation and Experiment

Despite prior reports indicating that end association becomes difficultas chain length increases ([62], [63], [64]), the regime of longtelechelic polymers (LTPs, FIG. 47A, right; see Table 13 for list ofpolymers) at concentrations below 1% was ventured into, theory was usedto guide the selection of molecular structures. To aid material design,a lattice model was used in which the polymer molecular weight simplymaps onto the corresponding number of connected lattice sites, each sitewith a volume equal to that of an effectively freely-jointed segment(“Kuhn segment”) a well-established property, tabulated for manypolymers (Table 14). Unsaturated hydrocarbon backbones were chosen basedon their solubility (remaining in solution down to the freezing point offuel) and strength (FIG. 78 see [9]). In addition to the Kuhn segmentvolume, two additional attributes of the polymer backbone enter into theentropy cost of ring closure: the Kuhn segment length (how close endsare for a ring to close) and the excluded volume parameter (how expandedthe chain is in solution). The end-association strength (i.e., energypenalty for unpaired ends) enters through the chemical potential of thelinear species.

To guide the design of an experimental system, the relationship of modelparameters to polymers having unsaturated hydrocarbon backbones1,4-polyisoprene (PI), 1,4-polybutadiene (PB) and polycyclooctadiene(PCOD) in Jet-A solvent is considered. The model is formulated withsites of volume a³ on a lattice with coordination number c=6. Thelattice size a³=v_(K), where v_(K)=MW_(K)/(N_(A)ρ) is the volume of aKuhn segment (with N_(A) denoting Avogadro's number and ρ, the polymerdensity) for a specific polymer of interest. A chain of molecular weightMW_(p) maps onto M=MW_(p)/MW_(K) connected lattice sites. To model asolution at volume fraction ϕ_(p), a system of Np polymer molecules in avolume V=(MN_(p)+N_(s))a³ is treated, the number of solvent latticesites adjusted to give the specified concentration, i.e.,N_(s)=MN_(p)(1−ϕ_(p))/ϕ_(p). To quantify the entropic cost of loopclosure (Example 47), numerical values are needed for the smallend-to-end distance x required to close a loop and for the number ofmonomers in a thermal blob g_(T)≈b⁶/v². For simplicity x/b=1 is chosen.

All three backbones of interest here, PI, PB and PCOD, can berepresented to good approximation by a single set of parameters, becausethe differences among them are relatively small (Table 14). For theseunsaturated hydrocarbon backbones, variations in molecular parametersresult from differences in cis/trans ratio of the backbone double bondsand the fraction of monomer insertions that create short side chains(3,4- and 1,2-units in PI and PB). If all six chain microstructures inTable 14 are considered, the lattice size is a=0.61±0.03 nm. Focus isplaced on polymers that have very few side chains (PCOD does not haveany); if only microstructures with ≤7% 3,4- and 1,2-units, the latticesize shifts very slightly to a=0.59±0.03 nm. Similarly, the molecularweight of a Kuhn segment is MW_(K)=121±22 g/mol if all sixmicrostructures are included and shifts slightly to MW_(K)=113+16 g/molif microstructures with 18% or more 3,4- and 1,2-units are excluded. TheKuhn step length is roughly 50% greater than the lattice size a: if allsix microstructures are included b=0.93±0.07 nm (with ≤7% 3,4- and1,2-units, b=0.90±0.08 nm). The excluded volume parameter v wasestimated as v/b³≈0.10 for PI in Jet-A, consistent with g_(T)˜100. [24]Based on literature results in cyclohexane, the expanded conformationsof PI and PB are very similar when they are dissolved in a good solventthat is similar to Jet-A (R_(g) [nm]=A(MW), with A=0.0129 for PB and0.0126 for PI, and with v=0.609 for PB and 0.610 for PI). Thus,theoretical predictions with a single set of parameters are expected toprovide equally good guidance for molecular design with PI, PB and PCODbackbones. Results are shown for MW_(K)=113 g/mol, a=0.61 nm, b=0.90 nm,x/b=1 and gT=100

The model predicts the equilibrium distribution of aggregates in termsof concentrations of supramolecular species with various sizes asfunctions of polymer concentration, length of the telechelic buildingblocks and binding energy. The model provides a guideline to achieve thedesired rheological benefits (mist control and drag reduction).

To overcome the problem of chain collapse that occurs when stickers aredistributed along a chain, a model of ring-chain equilibrium is used totest the hypothesis that clustering stickers at the ends of polymerchains can be used to generate a sufficient population ofmega-supramolecules to exert mist control. It is shown that linearchains displaying strongly associating, complementary end-groups (A-Aand B-B binary mixtures) form linear and cyclic supramolecules thatextend to “mega-supramolecules” if the individual building blocks arelong enough. Specifically, >50 ppm of mega-supramolecules (ca. 5-10×10⁶g/mol) is only achieved with telechelic chains of length >5×10⁵ g/moland a specific range of association energy, 16≤ε≤18. These calculationresults help to inspire the development of synthetic routes to the novelmolecules.

Experimentally, poly(1,5-cyclooctadiene) is chosen as an exemplarypolymer backbone for testing, which corresponds to a 1,4-polybutadienewith 75% cis, 25% trans and 0% short side branches. Thus, the requiredchain length for PCOD can be similar to that predicted with parametersbased on PI and PB as discussed in Example 39 taking into account thefact that unlike the model, real polymers are polydisperse. Thetelechelics synthesized using ROMP/CTA have M_(w)/M_(n)=1.5±0.1, so theguidance from theory is applied by targeting polymers in range fromM_(w)=5×10⁵ g/mol to M_(n)=5×10⁵ g/mol (M_(w)=10×10⁵ g/mol). Theremarkable polymers described in the paper demonstrate the success ofthe described theoretical model and the parameter estimation using priorliterature on 1,4-PI and 1,4-PB. Exemplary effects on the distributionof the polymers of parameters such as concentration, lengths of polymersand energy association (Ka) are reported in Examples 39 to 41 below.

Example 40: Effect of Concentration

A comparison of model results for MW_(p)=10⁶ g/mol (labeled 1000k) attotal polymer volume fraction ϕ_(total) of 1400 ppm and 800 ppm wasperformed. The models were obtained with the computation and experimentsof Example 38.

The results illustrated in FIG. 51 demonstrate two important effects oftotal polymer concentration. First, at fixed MW_(p) and ε, increasingconcentration improves the fraction of the polymer involved in largerlinear aggregates (compare upper right and middle right in FIG. 51): at1400 ppm the distribution of linear supramolecules (open symbols) decaysmore gradually with increasing M_(W), and the position of the peak inϕ_(linear) vs. M_(W) is greater at 1400 ppm than at 800 ppm (mostvisibly for ε=18, right column of FIG. 51).

Example 41: Effect of Length of Telechelic Building Blocks

Effects of polymer lengths on the polymer distribution of thepoly(1,5-cyclooctadiene in a host composition was determined by themodeling computation and experiments of Example 38.

Longer chains begin to overlap at a lower concentration than shorterones. To examine the effect of the length of the individual buildingblocks (MW_(p)) at similar degree of overlap, they are compared atϕ_(total)=¼ϕ* for their respective ϕ*(MW_(p)): results for 5×10⁵ g/molchains at 1400 ppm (bottom row, FIG. 51) and 1×10⁶ g/mol chains at 800ppm (middle row). The shape of the ϕ_(linear) vs. M_(w) does not changesignificantly with MW_(p) (for all E). An effect of MW_(p) is thatlonger telechelics reduce the fraction of polymer “wasted” in rings withsmall aggregation numbers, due to the increased entropic cost ofcyclization for larger loops.

Example 42: Effect of Energy of Association

Effects of energy of association on the polymer distribution of thepoly(1,5-cyclooctadiene in a host composition was determined by modelingcomputation and experiments of Example 38.

The equilibrium distribution changes qualitatively as the associationenergy increases (FIG. 51, from left to right): the population of loopsof all sizes increases (due to higher penalty for dangling ends) and thebreadth of the distribution of linear species broadens and the peak inϕ_(linear) decreases. At values of ε≤14, aggregates are few and thedominant components are the telechelic building blocks themselves. Atvalues of ε≥20 (not shown), the dominant components are cycles of lowM_(W). Intermediate values of the energy of association, correspondingto 16≤ε≤18, provide a balance of interactions strong enough to driveformation of large supramolecules and weak enough to accommodate asignificant population of linear superchains (with unpaired ends).

Example 43: Mist-Control Applications

The model showed that optimal formation of “mega-supramolecules” (linearsupramolecules of M_(W)≥5×10⁶ g/mol and cycles of M_(W)≥10×10⁶ g/mol)correlates with maximizing the equilibrium fraction of polymers involvedin linear supramolecules in the 5-10×10⁶ g/mol range.

Two key features of the distributions that satisfy this objective are(i) favorable partitioning of the polymer into linear rather than cyclicsupramolecules, and (ii) a well-defined peak in ϕ_(linear) centeredaround ˜5×10⁶ g/mol.

As expected, partitioning of the polymer into linear supramolecules isfavored at higher values of MW_(p) and ϕ_(total)—but both of thesequantities are constrained due to the limitations of shear degradation(M_(W)≤106 g/mol) and system compatibility for fuel (ϕ_(total)≤¼ϕ*).Near these maximal values, the strong dependence of the supramoleculardistributions on the energy of interactions has important implicationsfor mist-control applications. For mixtures of A-A and B-B molecules,model predictions indicate that favorable results will be found in arelatively narrow range of association energy, 16≤ε≤18.

Example 44: Model to Determine Lifetime of Equilibrium Distribution

The model assumes that under conditions of practical importance,equilibrium is restored as fast as it is disturbed. Therefore, the timetaken by a polymer to reach the equilibrium partitioning of the polymerinto aggregates of all sizes was investigated

The average lifetime of a donor-acceptor physical bond is estimatedusing τ_(b)·τ₀ exp(ε), where τ₀˜ηb³/kT describes a typical motional timefor a monomer in solvent with shear viscosity η. For solvents like fuel,η˜1 mPa·s, giving τ₀˜10¹⁰ s, so the lifetime is on the order of mv 1 msfor ε=17. Therefore, if equilibrium can be reached with roughly 10³bond-breaking and bond-forming events and for end-groups with 16-18 kTenergy of association, that time is on the order of is.

Example 45: Theoretical Treatment of Equilibrium Distribution of Cyclicand Linear Supramolecules

The inventory of all cyclic and linear supramolecules was computed withthe modeling of Example 43, as a function of concentration, backbonelength and end-association strength by solving the system of equilibriumrelationships in a population balance model (FIG. 47B). The resultingpredictions indicate that an adequate concentration ofmega-supramolecules (e.g., >50 ppm of supramolecules with M_(w)≥5,000kg/mol ([7])) form if the concentration of LTPs is 1,400 ppm, theirbackbone has approximately 6,000 Kuhn segments (M_(w)=500 kg/mol forpolycyclooctadiene, PCOD) and their ends associate pairwise with anenergy of 16-18 kT (modeling, after Goldstein([65]), FIGS. 51-56).Furthermore, theory shows that the favorable window of chain lengths andassociation strengths is relatively narrow. If the backbone is too short(e.g., 200 kg/mol PCOD), the fraction of material that is “lost” to theformation of small cyclics increases and, consequently, theconcentration of telechelics can be increased. If the backbone is toolong (e.g., 1,000 kg/mol PCOD), the individual telechelics becomesusceptible to degradation in strong flows (below). If the associationenergy is too low (e.g., 14 kT), formation of supramolecules isinadequate. If the association energy is too high (e.g., 20 kT),dangling ends are overly penalized and too few linear species form.

While there are already many studies of the theory of ring-chainequilibrium, of interest is formulating the problem so that problem ofequilibrium population as a function of the length, concentration andassociation energy could be readily solved. In the present construction,terms arising from microscopic interactions, as well as terms arisingfrom the center-of-mass and configurational entropy (except loopclosure) of polymer components and solvent in solution are carried outexplicitly. Whereas terms arising from (i) the energy of association ofthe end-groups within a polymer aggregate, and (ii) the entropic cost ofloop closure for cyclic supramolecular aggregates are absorbed into thestandard chemical potentials of the polymeric species.

As a first step in modeling the equilibrium distribution of cyclic andlinear supramolecules from telechelic polymers A-A and B-B (FIG. 52),the case of association of telechelic polymers A₁-A₂ and B₁-B₂ isstarted with. Subsequently, it is shown that the predicted distributionshold for A-A and B-B as well (Example 48). It is assumed that theend-groups A₁ and A₂, and likewise B₁ and B₂, are distinguishable but ofidentical reactivity (as though one end were isotopically labeled).

Example 46: Model Description: Equilibrium Using a Lattice Model

This equilibrium is approximated using a lattice model followingGoldstein [65]. A solution of N_(s) solvent molecules and N_(Atotal) andN_(Btotal) telechelic A₁-A₂ and B₁-B₂ chains of MA and MB repeat units,respectively, occupies a volume V that is partitioned into lattice sitesof volume a³, which is the volume of a solvent molecule and also thevolume of a monomer. There is negligible volume change upon mixing, so Va³(N_(s)+N_(Atotal)M_(A)+N_(Btotal)M_(B))=Λa³, where A is the totalnumber of “sites.” Subscripts i (or j) refer to polymeric components.Component i is composed of n_(i) A₁-A₂ building blocks and m_(i) B₁-B₂building blocks, and has M=n_(i)M_(A)+m_(i)/M_(B) repeat units. Thevolume fraction of solvent is ϕ_(s)=N_(s)/Λ and that of component i isϕ_(i)=N_(i)M_(i)/Λ. Unless otherwise specified, sums j, are over allpolymer components in solution, e.g., the sum of the volume fractions ofall polymeric species are equal the total polymer volume fractionϕ=Σ_(i)M_(i)N_(i)/Λ=1−ϕ.

The total free energy F of the solution is the sum of entropic andenthalpic contributions, F_(S) and F_(int), and of contributions fromthe internal free energy of solvent and polymer components:

$\begin{matrix}{F = {F_{int} + F_{S} + {N_{s}\mu_{s}^{0}} + {\sum\limits_{j}{N_{j}\mu_{j}^{0}}}}} & (1)\end{matrix}$

where μ_(f) ⁰ is the standard chemical potential of polymeric componentj. The first term is due to solvent-solvent, polymer-solvent, andpolymer-polymer interactions, which are estimated by the random mixingapproximation:

F _(int)=Λδ[(1−ϕ)² h _(ss)+ϕ² h _(pp)+2ϕ(1−ϕ)h _(ps)]  (2)

where δ is one-half the local coordination number, and h_(ij) are themicroscopic interaction energies of the polymer and solvent species. Thesecond term is due to configurational and center-of-mass entropy, S:

$\begin{matrix}{S = {{k{\sum\limits_{j}{\ln\;{\Omega\left( {0,N_{j}} \right)}}}} + {\Delta S_{mix}}}} & (3)\end{matrix}$

where Ω(0,N_(j)) is the number of possible configurations of N_(j)molecules of polymer component j each having M_(j) repeat units, ontoM_(j)N_(j) sites (i.e., pure component j before mixing with otherpolymer species or solvent). Following the notation of Hill [66] for theentropy of a melt of N_(i) linear polymer chains of length M_(i):

$\begin{matrix}{{\ln{\Omega\left( {0,N_{i}} \right)}} = {{{- N_{i}}\;\ln\; N_{i}} + N_{i} + {M_{i}N_{i}{\ln\left( {M_{i}N_{i}} \right)}} - {M_{i}N_{i}} + {{N_{i}\left( {M_{i} - 1} \right)}{\ln\left( \frac{c - 1}{M_{i}N_{i}} \right)}}}} & (4)\end{matrix}$

where c is the coordination number. The entropy of mixing of the solventand all polymer components, ΔS_(mix), is approximated using theFlory-Huggins expression:

$\begin{matrix}{{\Delta\; S_{mix}} = {- {{k\left( {{N_{s}\ln\;\phi_{s}} + {\sum\limits_{j}{N_{j}\ \ln\;\phi_{j}}}} \right)}.}}} & (5)\end{matrix}$

The entropic contribution to the free energy of the mixture istherefore:

$\begin{matrix}\begin{matrix}{F_{s} = {{{- T}\;\Delta\; S_{mix}} - {{kT}{\sum\limits_{j}{\ln\;{\Omega\left( {0,N_{j}} \right)}}}}}} \\{= {{{kT}\left\lbrack {{N_{s}{\ln\left( \frac{N_{s}}{\Lambda} \right)}} + {\sum\limits_{j}{N_{j}{\ln\left( \frac{N_{j}}{\Lambda} \right)}}}} \right\rbrack} +}} \\{{{kT}{\sum\limits_{j}{N_{j}{\ln\left( M_{j} \right)}}}} - {{kT}{\sum\limits_{j}{\ln\;{\Omega\left( {0,N_{j}} \right)}}}}}\end{matrix} & (6)\end{matrix}$

where ϕ_(s) and ϕ_(j) are replaced by N_(s)/Λ and N_(j)M_(j)/Λrespectively.

The equilibrium distribution of species is readily analyzed in terms ofthe chemical potentials of the solvent p and the polymeric species pi.For example, at equilibrium, the chemical potential of a supramolecularcomponent i made up of n_(i) A₁-A₂ and m_(i) B₁-B₂ satisfy theequilibrium condition:

μ_(i) =n _(i)μ_(A) +m _(i) U _(B)  (7)

where μ_(A) and μ_(B) are the chemical potentials of building blocksA₁-A₂ and B1-B₂, respectively. The chemical potential of polymercomponent i involves both interactions (solvent-solvent, polymer-solventand polymer-polymer) and entropic contributions. The contribution to thechemical potential of component i due to interactions is:

$\begin{matrix}{{{\mu_{{int},i} = \frac{\partial F_{int}}{\partial N_{i}}}}_{N_{j \neq i}} = {{{- \omega}\; M_{i}\phi_{s}^{2}} + {\omega_{pp}M_{i}}}} & (8)\end{matrix}$

where ϕ=(M_(i)N_(i)+Σ_(j≠i)M_(j)N_(j))/Λ withΛ=N_(s)+M_(i)N_(i)+Σ_(j≠i)M^(j)N_(j) and ϕ_(s)=1−ϕ are used and, forconvenience, ω_(mn)=δh_(mn) and ω=ω_(pp)+ω_(ss)−2ω_(ps) are introduced.The entropic contribution to the chemical potential of component i is:

$\begin{matrix}{{{\frac{\mu_{S,i}}{kT} = {\frac{1}{kT}\frac{\partial F_{S}}{\partial N_{i}}}}}_{N_{j \neq}} = {{\ln\left( \frac{\phi_{i}}{M_{i}} \right)} + 1 - \phi_{i} - {M_{i}\left\lbrack {\phi_{s} + {\sum\limits_{j \neq i}\frac{\phi_{j}}{M_{j}}}} \right\rbrack} + {\ln M_{i}} - 1 - {M_{i}\left\lbrack {{\ln\left( {c - 1} \right)} - 1} \right\rbrack} - {\ln M_{i}} + {{\ln\left( {c - 1} \right)}.}}} & (9)\end{matrix}$

Differentiation of Equation 6 and substitution of Equations 7 and 9 givethe following expression for the chemical potential of component i,valid for the single-chain building blocks and all supramolecules:

$\begin{matrix}{{{\mu_{i} = \frac{\partial F}{\partial N_{i}}}}_{N_{j \neq i}} = {\mu_{i}^{0} + {kT\left\{ {{\ln\left( \frac{\phi_{i}}{M_{i}} \right)} - {M_{i}\left\lbrack {\phi_{s} + {\sum\limits_{j}\frac{\phi_{j}}{M_{j}}}} \right\rbrack} + f_{i}} \right\}} - {\omega\; M_{i}\phi_{s}^{2}} + {\omega_{pp}M_{i}}}} & (10)\end{matrix}$

where f_(i)=ln(c−1)+M_(i)[1−ln(c−1)]. Substituting the expressions forμ_(i), μ_(A), and μB from Equation 10 into Equation 7 above, afterrearrangement, the following mass-action relation for component l isobtained:

$\begin{matrix}{{\mu_{i}^{0} + {k{T\left\lbrack {{\ln\left( \frac{\phi_{i}}{M_{i}} \right)} + f_{i}} \right\rbrack}}} = {{n_{i}\mu_{A}^{0}} + {m_{i}\mu_{B}^{0}} + {k{T\left\lbrack {{n_{i}{\ln\left( \frac{\phi_{A}}{M_{A}} \right)}} + {m_{i}{\ln\left( \frac{\phi_{B}}{M_{B}} \right)}} + {n_{i}f_{A}} + {m_{i}f_{B}}} \right\rbrack}}}} & (11)\end{matrix}$

where ϕ_(A) and ϕ_(B) are the equilibrium volume fractions of the freetelechelics A₁-A₂ and B1-B₂, respectively. It is convenient to rewriteEquation 11 as follows:

$\begin{matrix}{{\left( \frac{\phi_{i}}{{n_{i}M_{A}} + {m_{i}M_{B}}} \right) = {\left( \frac{\phi_{A}}{M_{A}} \right)^{n_{i}}\left( \frac{\phi_{B}}{M_{B}} \right)^{m_{i}}{\exp\left( \Gamma_{i} \right)}}},} & (12) \\{{{where}\mspace{14mu}\Gamma_{i}} = {{\frac{1}{kT}\left( {{n_{i}\mu_{A}^{0}} + {m_{i}\mu_{B}^{0}} - \mu_{i}^{0}} \right)} + {\left( {n_{i} + m_{i} - 1} \right){{\ln\left( {c - 1} \right)}.}}}} & (13)\end{matrix}$

The conservation equations are then:

$\begin{matrix}{{\phi_{Atotal} = {{\sum\limits_{j}{\phi_{j}\left( \frac{n_{j}M_{A}}{{n_{j}M_{A}} + {m_{j}M_{B}}} \right)}} = {\sum\limits_{j}{n_{j}{M_{A}\left( \frac{\phi_{A}}{M_{A}} \right)}^{n_{j}}\left( \frac{\phi_{B}}{M_{B}} \right)^{m_{j}}{\exp\left( \Gamma_{j} \right)}}}}}{\phi_{Btotal} = {{\sum\limits_{j}{\phi_{j}\left( \frac{m_{j}M_{A}}{{n_{j}M_{A}} + {m_{j}M_{B}}} \right)}} = {\sum\limits_{j}{m_{j}{M_{B}\left( \frac{\phi_{A}}{M_{A}} \right)}^{n_{j}}\left( \frac{\phi_{B}}{M_{B}} \right)^{m_{j}}{{\exp\left( \Gamma_{j} \right)}.}}}}}} & (14)\end{matrix}$

To this point, the formulation has treated terms arising frommicroscopic interactions, as well as center-of-mass and configurationalentropy (except loop closure) of polymer components and solvent. Next(i) the energy of association of the paired end-groups within asupramolecule and (ii) the entropic cost of loop closure for cyclicsupramolecules are accounted for, which are incorporated into thestandard chemical potentials μ_(j) ⁰.

For this purpose, it is useful to identify “groups” of polymer species,each assigned an index g, that are topologically similar and have thesame values of M_(j)=M_(g), n_(j)=n_(g), m_(j)=m_(g), and Γ_(j)=Γ_(g).In identifying “groups” of polymer species, A and B are used to refer toA1 or A2 and B1 or B2, respectively (FIG. 53). In counting number ofdistinct species in group g (Ω_(g)) the two ends of an A-telechelic or aB-telechelic are treated as distinguishable. Thus, group g is composedof all the different possible aggregates obtained by the assembly of theA1-A2 and B1-B2 building blocks. For example, group g=3 has Ω_(g)=4distinct aggregates (FIG. 53): A1-A2-B1-B2, A1-A2-B2-B1, A2-A1-B1-B2,and A2-A1-B2-B1.

How many components belong to each group? For linear aggregates thereare two possibilities: (i) for n_(g)+m_(g) even (i.e., n_(g)=m_(g)), nosequence read from left to right will be the same as a sequence readfrom right to left, so the number of ways to arrange the molecules isΩ_(g)=2^(n) ^(g) ^(+m) ^(g) ; (ii) for n_(g)+m_(g) odd, every sequenceread from left to right will have a matching sequence read from right toleft, so the number of ways to arrange the molecules is Ω_(g)=2^(n) ^(g)^(+m) ^(g) ⁻¹. Supramolecular cycles always have n_(g)=m_(g). The numberof ways to form such a loop is derived in Example 48 below; to goodapproximation it is Ω_(cyc,g)=2+(2^(2n) ^(g) ⁻¹−2)/n_(g).

The fact that (by construction) all of the components j in anyparticular group g have the same value of μ_(j) ⁰=μ_(g) ⁰ allows theequilibrium condition and the conservation equations to be rewritten interms of ϕ_(g), the cumulative volume fraction of all polymer componentsin group g:

$\begin{matrix}{\left( \frac{\phi_{g}}{{n_{g}M_{g}} + {m_{g}M_{g}}} \right) = {{\Omega_{g}\left( \frac{\phi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\phi_{B}}{M_{B}} \right)^{m_{g}}{\exp\left( \Gamma_{g} \right)}}} & (15) \\{{\phi_{Atotal} = {\sum\limits_{g}{n_{g}M_{A}{\Omega_{g}\left( \frac{\phi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\phi_{B}}{M_{B}} \right)^{m_{g}}\exp\left( \Gamma_{g} \right)}}}{\phi_{Btotal} = {\sum\limits_{g}{m_{g}M_{B}{\Omega_{g}\left( \frac{\phi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\phi_{B}}{M_{B}} \right)^{m_{g}}{\exp\left( \Gamma_{g} \right)}}}}} & (16)\end{matrix}$

The standard chemical potentials μ_(g) ⁰ include the appropriatemultiples of the standard chemical potentials of the A-A and B-Bbuilding blocks and the appropriate multiple of the association energyεkT. For a cyclic group, there is an additional term due to the entropycost of ring closure, ΔS_(loop)=−k ln G_(cyc), where G_(cyc) is theprobability density (treated in Example 47 below) for closure of a groupg ring:

$\begin{matrix}{\mu_{g}^{0} = \left\{ \begin{matrix}{{n_{g}\mu_{A}^{0}} + {m_{g}\mu_{B}^{0}} - {ɛ\;{{kT}\left( {n_{g} + m_{g}} \right)}} - {kT\ln G_{{cycl},g}}} & {{if}\mspace{14mu}{cyclic}} \\{{n_{g}\mu_{A}^{0}} + {m_{g}\mu_{B}^{0}} - {ɛ\;{{kT}\left( {n_{g} + m_{g} - 1} \right)}}} & {{{if}\mspace{14mu}{linear}},}\end{matrix} \right.} & (17)\end{matrix}$

so that Γ_(g) in the equilibrium and conservation relationships(Equations 15 and 16) is:

$\begin{matrix}{\Gamma_{g} = \left\{ \begin{matrix}{{ɛ\left( {n_{g} + m_{i}} \right)} + {\left( {n_{i} + m_{i} - 1} \right){\ln\left( {c - 1} \right)}} + {\ln G_{{cycl},g}}} & {{if}\mspace{14mu}{cyclic}} \\{{ɛ\left( {n_{g} + m_{g} - 1} \right)} + {\left( {n_{g} + m_{g} - 1} \right){\ln\left( {c - 1} \right)}}} & {{if}\mspace{14mu}{{linear}.}}\end{matrix} \right.} & (18)\end{matrix}$

Example 47: Entropic Cost of Loop Closure

The entropic cost of loop closure is determined by calculating theprobability of loop closure, as follows: For Gaussian linear chains of NKuhn monomers of length b, the probability density function for theend-to-end vector r is [24]:

$\begin{matrix}{{G_{Gaussian}\left( {r,N} \right)} = {\left( \frac{3}{2\pi\;{Nb}^{2}} \right)^{\frac{3}{2}}\exp{\left\{ {- \frac{3r^{2}}{2Nb^{2}}} \right\}.}}} & (19)\end{matrix}$

The argument within the exponential −3r²/(2Nb²)≅0 for ∥r∥<<r²>^(1/2), sothe probability that the chain ends be within a small distance x of eachother, where x/b˜O(1), is:

$\begin{matrix}{G_{{cyc},{Gaussian}} = {{\left( \frac{3}{2\pi\;{Nb}^{2}} \right)^{\frac{3}{2}}{\underset{0}{\int\limits^{2\pi}}{d\;\phi{\underset{0}{\int\limits^{\pi}}{d\;{\theta sin}\;\theta{\underset{0}{\int\limits^{x/b}}{{{dr} \cdot r^{2}}{\exp(0)}}}}}}}} = {{4{\pi\left( \frac{3}{2\pi\;{Nb}^{2}} \right)}^{\frac{3}{2}}{\underset{0}{\int\limits^{x/b}}{{{dr} \cdot r^{2}}{\exp(0)}}}} = {\left( \frac{6}{\pi\; N^{3}} \right)^{\frac{1}{2}}{\left( \frac{x}{b} \right)^{3}.}}}}} & (20)\end{matrix}$

For real chains, excluded volume interactions of the monomers at chainends reduce the probability density function G(r,N) by the factor

$\begin{matrix}{{\frac{G_{real}\left( {r,N} \right)}{G_{Gaussian}\left( {r,N} \right)}\text{∼}\left( \frac{r}{\sqrt{\left\langle r^{2} \right\rangle}} \right)^{\gamma}\mspace{14mu}{for}\mspace{14mu}\frac{r}{\sqrt{\left\langle r^{2} \right\rangle}}} ⪡ 1} & (21)\end{matrix}$

where the exponent γ≅0.28 [24], so that the probability of cyclizationbecomes

$\begin{matrix}{G_{{cyc},{real}} \approx {4{\pi\left( \frac{3}{2\pi\;{Nb}^{2}} \right)}^{\frac{3}{2}}\left( \frac{1}{{bN}^{v}} \right)^{\gamma}{\underset{0}{\int\limits^{x/b}}{{{dr} \cdot r^{2 + g}}{\exp(0)}\text{\textasciitilde}N^{{{- 3}/2} - {\gamma\; v}}}}}} & (22)\end{matrix}$

where the fractal exponent v is 0.588 in good solvent. The loop closureprobability thus scales as N^(−3/2) for Gaussian chains and N^(−1.66)for swollen chains. The entropic cost of loop closure is simplyΔS_(loop)=−k ln G_(cyc).

In dilute or semi-dilute solutions, all chain segments smaller than thethermal blob g_(T)≈b⁶/v² (where v is the excluded volume parameter) haveGaussian statistics because excluded volume interactions are weaker thanthe thermal energy. At the concentrations of interest, the total polymervolume fraction ϕ=Σ_(j)ϕ_(j) is low enough to ignore polymer-polymeroverlap, so the following expression is appropriate for the entropiccost of loop closure ΔS_(loop)=−k ln G_(cyc) for any cyclic aggregatej:

$\begin{matrix}{G_{{cyc},j} \approx {\left( \frac{6}{\pi g_{T}^{3}} \right)^{\frac{1}{2}}\left( \frac{x}{b} \right)^{3}{\left( \frac{M_{j}}{g_{T}} \right)^{- 1.66}.}}} & (23)\end{matrix}$

That is, all chain segments larger than gr are fully swollen.

Example 48: Number of Ways to Form Loops

To determine the number of different loops that can be formed by linkingn A-A and n B-B telechelic chains end-to-end via association of A and Bend-groups (FIG. 53 left), telechelics are started to be treated withdistinguishable ends (i.e., n A₁-A₂ molecules that are indistinguishablefrom each other, and likewise n B₁-B₂ molecules). This way of treatmentmaps onto the combinatorial problem of counting necklaces formed usingbeads of different colors, in which two necklaces are consideredequivalent if one can be rotated to give the other. By viewing eachsupramolecular loop in terms of adjacent pairs of telechelics (with oneA₁-A₂ and one B₁-B₂ molecule per pair), they correspond to necklacesmade up of n “beads” of 4 “colors” (FIG. 54). For example,A1A2B1B2=black, A1A2B2B1=white, A2A1B1B2=blue, and A2A1B2B1=green can bechosen. The formula for the number of different necklaces is [67]:

$\begin{matrix}{{m(n)} = {\frac{1}{n}{\sum\limits_{d❘n}\left\lbrack {{\varphi(d)} \cdot 4^{n/d}} \right\rbrack}}} & (24)\end{matrix}$

where the sum is over all numbers d that divide n, and p(d) is the Eulerphi function.

In reality, the above formula overcounts the number of ways to formsupramolecular loops by a factor of 2. The number of distinct cyclicsupramolecules s(n) in the set obtained from n A₁-A₂ and n B₁-B₂telechelic chains, {loops_(n)}, can be seen to be half the number ofdistinct necklaces of n beads of four colors {necklaces_(n)} because anysupramolecular loop “reads” as a distinct necklace clockwise vs.counter-clockwise (FIG. 55). While each necklace in {necklaces_(n)}uniquely maps onto a supramolecular loop in {loopsn}, every loop in{loops_(n)} maps back to two different necklaces, which belong to{necklaces_(n)}. The elements of {necklaces_(n)} can be arrangedpairwise, revealing that there are twice as many elements in{necklaces_(n)} as in {loops_(n)}. Therefore, the number of distinctsupramolecular loops s(n) is:

$\begin{matrix}{{s(n)} = {\frac{1}{2n}{\sum\limits_{d|n}\left\lbrack {{\varphi(d)} \cdot 4^{n/d}} \right\rbrack}}} & (25)\end{matrix}$

To see that the result obtained by treating the end groups asdistinguishable gives the correct result for the actual case in whichA-ends are indistinguishable and likewise for B-ends, the reversibleassociation reactions are considered in FIG. 56. The reverse reactionrates are all identical. However, the forward reaction for case a(monotelechelic chains) is clearly one-fourth that of case b(telechelics with indistinguishable end-groups). In case c, there arefour identical intramolecular scission reactions that give the startingproducts, so the forward reaction in case c can be 4-fold faster thanthat of the forward reaction in case a. Thus, the difference in thenumber of ways to form dimers (Ω_(c)=4 compared to Ω_(a)=1) can be usedto evaluate the increased contact probability of the end-groups to formthe product. If the end-groups A, A₁, and A₂ have precisely the samereactivity, and likewise the end-groups B, B₁, and B₂, there cannot beany difference in the equilibrium partitioning of the molecules in casesb and c. This argument is generalized to conclude that the solution tothe equilibrium problem presented in FIG. 52, where end-groups areindistinguishable, is the solution which is developed for telechelicsA₁-A₂ and B₁-B₂, where end-groups are distinguishable. A less carefulmodeling of the association of telechelic polymers A-A and B-B mightmiscalculate the cumulative equilibrium volume fraction of polymeraggregates that fall within any group g by omitting the factor Q_(g) inEquation 15.

Example 49: Computation of Volume Fraction at Equilibrium

The following procedure was used to calculate the volume fraction of allpolymer components (i.e., single-chain starting materials and aggregatesof all sizes) at equilibrium, for polymer solutions of A₁-A₂ and B₁-B₂telechelics of specified molecular weights at specified initialconcentrations ϕ_(Atotal) and ϕ_(Btotal), (polymer components weregrouped as shown in FIG. 53):

First, a number of groups T_(groups) is chosen to include in theanalysis (even though there is an infinite number of possible polymercomponents, it is expected that above a certain size, polymer aggregateswill have negligible equilibrium volume fraction and can therefore beignored).

Calculate n_(g), m_(g), M_(g), Ω_(g), G_(cyc,g) (if appropriate), andΓ_(g) for polymer group g, for g=1 . . . T_(groups).

Solve the conservation equations, Equations 16, for (ϕ_(A), ϕ_(B)).

Calculate ϕ_(g) for g=1 . . . T_(groups) using Equation 15.

Repeat with a new value of T_(groups) twice that of the previous oneuntil changes in the calculated values of ϕ_(g) from one value ofT_(groups) to the next are negligible.

Example 50: Selection of End-Groups

FIG. 57, Panel A shows the chemical structures and molar masses of theend-associative polymers (excepting isophthalic acid/tertiary aminefunctionalized ones that are shown in FIG. 47C). FIG. 57, Panel B showsthe specific viscosities of telechelic polymers at 8.7 mg/ml totalpolymer in 1-chlorododecane. Based on the literature on complementarypolyvalent hydrogen-bonding pairs, it is shown that a 1:1 THY/DAAPsolution had a viscosity equal to the average of the viscosities of theindividual components' solutions. It is also shown that when the 1:1HR/CA showed a viscosity equal to the average of the individualcomponents. Only the DA/DB pair shows enhancement in viscosity relativeto the individual telechelic polymers. FIG. 57, Panel C illustrates thesecondary electrostatic interactions (SEIs) in THY/DAAP and HR/CA pair.

The data suggest that, despite the simplicity of carboxylic acid andtertiary amine structures, the DA/DB pair provides strongerend-association than the hexadentate HR/CA pair. This difference isprimarily attributed to the 3- to 4-fold greater strength ofcharge-assisted hydrogen bonds (as is the case of DA/DB) relative toordinary hydrogen bonds (in both THY/DAAP and HR/CA). Therefore, innon-polar solvent the sum of the two charge-assisted hydrogen-bonds in aDA/DB pair is likely stronger than the sum of the six ordinary hydrogenbonds in the HR/CA pair. In addition, the DA/DB pair does not sufferfrom the adverse effect of repulsive secondary electrostaticinteractions (SEIs) that occur when the both partners have H-bond donorsand H-bond acceptors: in the HR/CA pair, the polarities of the sixhydrogen-bonds alternate in direction, thus decreasing the overallstrength of HR/CA association. It is estimated that for THY/DAAP(association constant in deuterated chloroform at 25° C.=10⁴ M⁻¹) [37],three primary hydrogen bonds contribute −24 kJ/mol and four repulsiveSEIs contribute+12 kJ/mol (net ca. 5 kT); and for HR/CA, six hydrogenbonds contribute −47 kJ/mol and eight repulsive SEIs contribute+23kJ/mol (net ca. 10 kT, FIG. 57, Panel C). The literature value of theassociation constant for a polymer-bound HR/CA pair in deuteratedchloroform at 25° C. is 1.5×10⁴ M⁻¹ (7), corresponding to anend-association strength of 9.6 kT, in good agreement with the value of10 kT estimated from SEI analysis. As described herein, the strength ofthe DA/DB pair is estimated to be 16-18 kT. The difference in estimatedstrength between DA/DB and HR/CA is consistent with the disclosedexperimental results in FIG. 57, Panel B. Together, SEI analysis andshear viscometry reveal that HR/CA does not, in fact, have anassociation constant in non-polar solvents that is high enough to drivelong telechelic polymers to form mega-supramolecules at concentrationsof interest in the scope of the present work.

Example 51: ¹H NMR Study of Incorporation of Chain Transfer Agent (CTA)into Polymer

To install functional groups at both chain ends with high fidelity(>95%, FIGS. 58-59), a two-step ring-opening metathesis polymerization(ROMP) protocol (FIG. 47C) ([68], [69]) in the presence of a chaintransfer agent (CTA) is used. Polymers conforming to the theory aresynthesized using carefully purified cis,cis-1,5-cyclooctadiene (COD,FIG. 60 ([69], [70])) and CTAs bearing functional end groups (ratio ofCOD:CTA>3,000:1, adjusted to give the desired molecular weight). Endgroups with discrete numbers of hydrogen bonds (di-functional ends,denoted DA/DB and tetra-functional ends, denoted TA) (FIG. 47C) can beinstalled after polymerization by conversion of ester- or chloride-endedpolymers (which serve as non-associative controls, NA), with degrees ofconversion >95% (FIGS. 61A-62C). To test predicted effects of backbonelength, corresponding telechelics with shorter backbones (e.g., FIG.48A, M_(w)˜45, 140, 300 kg/mol, see Table 13) were prepared.

FIGS. 58 and 59 show incorporation of CTA into polymer during the firststage of two-stage ROMP of COD, and chain extension to long telechelicsin the second stage. FIG. 58 ¹H NMR of characteristic peaks fordi(di-tert-butyl-isophthalate) CTA (structure of end-group shown inFIGS. 61A and 61B), unreacted CTA (proton 1) and CTA incorporated intomacromer (proton 2), at three time points; the integrations of the peakswere used to calculate the percentage of unreacted CTA, shown in partFIG. 59A. FIG. 59A, Kinetic curves show that the peaks characteristic ofthe unincorporated CTA are already difficult to quantify in the sampletaken after 40 min, and it is not evident for the sample taken at 1 hour(given the magnitude of the noise in the spectra, the amount ofunincorporated CTA is less than 3%). Dashed curve is calculated basedthe data point at 10 min assuming exponential decay of unreacted CTA.FIG. 59B, In an example with di-chloro PCOD, the M_(n) calculated by NMRis in good agreement with that measured by GPC, considering the inherentuncertainty in NMR integration and the inherent uncertainty in GPCmeasurement (5-10%). FIG. 59C, GPC traces show no indication of macroCTA (42 kg/mol) in the chain-extended telechelics (structure shown in D,497 kg/mol) produced in the second step.

Example 52: Conversion of Non-Associative (NA) End-Groups to AssociativeEnd-Groups

While theoretical predictions identify a class of polymers promising asmist-control additives for kerosene, telechelics of the length required(M_(w)>400 kg/mol, M_(w)/M_(n) ca. 1.5), in reality, are unprecedented.In order to test the predictions regarding such telechelic polymers, atwo-step ring-opening metathesis polymerization (ROMP) protocol in thepresence of a chain transfer agent (CTA) is adopted, as reports indicateit could produce relatively long telechelics with M_(w) up to ca. 260kg/mol (FIG. 45, Panels A-B and FIG. 47C).[68, 71] Cyclooctadiene (COD)is selected as the monomer because it has an adequate ring strain todrive ROMP and provides a backbone that has both strength and solubilityin hydrocarbons.[9, 72] Once carefully purified COD is used, telechelicsof the required length (M_(w)>400 kg/mol, up to 1,000 kg/mol if desired)and end functionality (>95%) are accessible.

Associative groups of interest can be installed at both ends of eachpolymer with high fidelity using custom CTAs, a built-in benefit of theROMP chemistry. In hydrocarbons, end-group association bycharge-assisted hydrogen bonding (such as carboxylic acid/tertiary amineinteraction) is particularly effective for building supramolecules.[73]Hence, in this study well-defined end-groups with discrete numbers ofhydrogen bonds are synthesized: isophthalic acid and di(tertiary amine)(denoted DA/DB for diacid/dibase), and di(isophthalic acid) andtetra(tertiary amine) (TA/TB) (FIG. 45, Panels A-B and FIG. 47C). Acidand amine end-groups are installed after polymerization by conversion ofester- or chloride-ended polymers (which serve as matchednon-associative negative controls, NA).

FIGS. 61A-61B show FIG. 61A, Structures of non-associative (NA)end-groups and the conversion from NA to associative end-groups: FIG.61B, isophthalic acid. FIG. 45, Panel A shows tertiary amine (productsshown in FIG. 47). Isophthalic acid end groups are obtained bydeprotection of the tBu groups in the tBu-ester-ended non-associativeprecursor. Tertiary amine end-groups are obtained via conversion ofchloride end-groups to azide end-groups, followed by an alkyne/azidecycloaddition.

Example 53: ¹H NMR Study of Degree of Conversion of the End-Groups

Conversion of tBu-ester to carboxylic acid as end-groups onpolycyclooctadiene is monitored by the peak for tBu group in the ¹H NMRspectra. FIGS. 62A and 62B show ¹H NMR spectra of tBu-ester ended (DE)and isophthalic acid ended (DA) polycyclooctadiene (M_(w)=630 kg/mol) toshow high degree of conversion of the end-groups. FIG. 62A the peaks forprotons on the phenyl ring (protons 1 and 2) shift due to the removal oftBu. Comparing the integration of peak for proton 2 (˜7.82 ppm) withthat of the baseline at ˜7.7 ppm (where the peak for proton 2 in DE is,see FIG. 62A top) in the spectrum of DA (FIG. 62A bottom) shows a <5% (1comparing to 0.04) potential unconverted end-groups due to baselinenoise. FIG. 62B the peak for tBu group disappears in the spectrum forDA, indicating removal of the tBu group.

Example 54: ¹H NMR Study of Azide Conversion to Tertiary Amine

Similarly, conversion of azide (obtained via conversion of chloride endgroups) to tertiary amine (obtained via an alkyne/azide cycloaddition,see FIG. 45, Panel B) as end-groups on polycyclooctadiene is monitoredby the proton peaks for triazole and phenyl rings in the ¹H NMR spectra.FIG. 62C ¹H NMR spectra of azide ended (DN₃) and tertiary amine ended(DB) polycyclooctadiene (M_(w)=540 kg/mol) to show high degree ofconversion of the end-groups. In the spectrum for DB (bottom), thepresence of a peak at 7.4 ppm indicates the formation of triazole rings(proton 5), absent in DN₃'s spectrum (top). The peak for protons on thephenyl ring (at positions 1 and 2) shifts from 6.85 ppm before (top) to6.75 ppm after the cycloaddition reaction (bottom): integration of thepeak for protons at 1 and 2 (˜6.75 ppm, relative integral integral=3) inthe spectrum of DB (bottom) and of the baseline at ˜6.85 ppm (nodetectable 1,2 of DN₃, relative integral=0.09) places an upper bound of<5% unconverted end-groups.

Example 55: Formation of Supramolecules and Effect of Excess TertiaryAmine

FIG. 63 shows formation of supramolecules in equimolar solutions ofα,ω-di(isophthalic acid) polycyclooctadiene, α,ω-di(di(tertiary amine))polycyclooctadiene (DA/DB), with non-associated controls (NA, see FIG.61A top; and solutions treated with an excess of a small-moleculetertiary amine, triethylamine, TEA at 10 μl/ml). FIG. 63, Panel A,Effect of chain length (k refers to kg/mol) on specific viscosity oftelechelics in tetralin and Jet-A (2 mg/ml) at 25° C. FIG. 63, Panel B,Effect of TEA (2.5 μl/ml) on the viscosities of associative telechelicpolymers DA/DB. FIG. 63, Panel C, Left: Static light scattering showsthat association between DA and DB chains (circle: 670k series;triangle: 300k series) in cyclohexane (CH) at 0.22 mg/ml (0.028%)produces supramolecules (filled), which separate into individualbuilding blocks (x) when an excess of a small-molecule tertiary amine isadded (open symbols, 10 μl/ml of triethylamine, TEA). Curves showpredictions of the model (see Examples 37-49)). Right: Zimm plot of thesame static light scattering data shown in Left part. Lines indicate thefitting to the Zimm equation and dashed lines indicate the extrapolationthat was used to evaluate the intercept at zero concentration, zeroangle; the slope of the line and the value of the intercept are used toevaluate the apparent M_(w) and apparent R_(g), details below. FIG. 63,Panel D, Resulting values of apparent M_(w) and R_(g) for the fivepolymer solutions in FIG. 63, Panel C.

The effect of chain length on specific viscosity of telechelics intetralin and Jet-A (FIG. 63, Panel A) is similar to that in cyclohexane(FIG. 48A). The specific viscosity of telechelics in Jet-A is generallylower than that in tetralin or cyclohexane. This effect is observed evenfor the non-associative polymers (NA), indicating that the backboneadopts a more compact conformation in Jet-A. This effect is related tothe composition of Jet-A as a mixture of many hydrocarbons with numberof carbon atoms between 6 and 16, including some components that aregood solvents for PCOD and some that are theta solvents for PCOD.

The model calculations FIG. 63, Panel C show the effect of doubling thebackbone length for complementary telechelics with association energy 16kT, backbone lengths corresponding to a PCOD of 1,000 kg/mol (x) or 500kg/mol (+) at 1,400 ppm concentration in a good solvent on thescattering pattern computed from the distribution of supramolecules(solid, supramolecules up to 9 telechelics; dashed, correspondingperfectly monodisperse non-associative telechelics). To compare with theexperimental data, a single vertical shift was allowed to be applied toall four curves and a single horizontal shift. The distributions ofsupramolecules are shown in FIG. 64.

The Zimm fitting was performed using Wyatt Astra Software (version5.3.4): illustrations for the 300k DA/DB and 300k DB are shown, with thelinear regression through the data (black solid line) extrapolated tozero-concentration (horizontal light gray dashed line) and to zero angle(oblique gray dashed line). The y-intercept of the zero-anglezero-concentration extrapolation gives the apparent M_(w), while itsslope is used to compute the apparent R_(g).

Example 56: Interplay of Telechelic Length and Concentration

Mega-supramolecules are formed at low concentration that behave likeultra-long polymers, exhibiting expanded (“self-avoiding”) conformationat rest and capable of high elongation under flow (FIG. 47A, right).This is in contrast to the collapsed, inextensible supramolecules formedby long chains with associative groups distributed along their backbone(FIG. 47A, left) ([74], [75]). To mimic ultra-long polymers, associationcan occur at chain ends and be predominantly pairwise. In contrast tomultimeric association ([62], [64]) that leads to flower-like micellesat low concentration (FIG. 47A, middle), recent studies have shown thatpairwise association is readily achieved for short chains with M_(w)≤50kg/mol using hydrogen bonding ([63], [76], [77], [78], [79], [80], [81],[82]). At low concentration, these have no significant rheologicaleffects, consistent with the theory of ring-chain equilibrium ([83],[84], [85], [86], [87]): small rings are the predominant species at lowconcentration (FIG. 47A, middle). It was realized that using very longchains as the building blocks would disfavor rings, because the entropycost of closing a ring increases strongly with chain length.

FIG. 64 shows modeling of interplay of telechelic length andconcentration in a stoichiometric mixture of complementaryend-associative telechelics in the regime of long telechelics(corresponding to ≥0.5 Mg/mol for high-1,4-polyisoprene,high-1,4-polybutadiene or polycyclooctadiene) and low concentration(≤0.14% wt/wt), facilitating comparison among the three different cases(FIG. 51, center column), in terms of both the number of telechelics ineach supramolecular species and the molecular weight of eachsupramolecular species. Symmetric cases are considered (donor andacceptor telechelics have the same length). End association energybetween donor and acceptor end-groups is 16 kT. The concentration ofeach distinct species is shown for supramolecules composed of up to 12telechelics; the symbol in a square outline represents the sum of allsupramolecules containing 13 or more telechelics (square around x is forthe case 1.0 Mg/mol chains at 1,400 ppm concentration; the squarearound + is for the other case in each graph). FIG. 64, Panel A, Effectof telechelic length on the distribution of the number of telechelics ina supramolecule, given as the concentration in ppm wt/wt of eachspecies, cyclic (circles) or linear (x or +), at a fixed totalconcentration of 1400 ppm. FIG. 64, Panel B, The same distributions asin A, presented in terms of the molar mass of the supramolecules; theweight-average molar mass of the supramolecules is given to the left ofthe legend. FIG. 64, Panel C, Effect of concentration on thedistribution of supramolecules for telechelics of 1M g/mol (hence, thenumber of telechelics in a given supramolecule is also its molar mass inMg/mol) Note the results for the 1 Mg/mol telechelics at 0.14%concentration is given in all three graphs to facilitate comparisons(see Examples 37-49).

In the regime of long telecheclics at low concentration, the equilibriumdistribution of rings is dominated by rings composed of 2 telechelics(one donor+one acceptor) or 4 telechelics (in a donor/acceptor system,rings can only close if the number of telechelics is even). The fractionof telechlics “lost” to these rings is cut in half by doubling thelength of the telechelics from 0.5M to 1.0 Mg/mol, increasing theformation of linear supramolecules FIG. 64, Panel A. Increasing thelength of the backbone also increases the size of the supramolecules ateach number of telechelics per supramolecules (compare FIG. 64, Panel Bto FIG. 64, Panel A); consequently, increasing the telechelic lengthstrongly increases the population of “mega-supramolecules” (the sum ofthe concentrations of all species having molecular weight greater than 5Mg/mol increases from 200 ppm for 0.5 Mg/mol telechelics to 400 ppm for1.0 Mg/mol). Dilution, here from 1,400 ppm to 800 ppm wt/wt, favors theformation of “small” supramolecules composed of 4 or fewer telechelicsat the expense of mega-supramolecules (here, the sum of all species >5Mg/mol falls from 400 ppm to 230 ppm). Note that “small” speciesassembled from 3-4 telechelics were already the dominant ones at higherconcentration, so dilution has relatively mild effects on the weightaverage molecular weight (numbers shown to the left of the legend inFIG. 64, Panel C). For further details on the model, please seemodeling.

Example 57: Shear Viscometry Study of LTPs with Donor-Acceptor TypeEnd-Groups

Shear viscometry study of donor-acceptor type LTPs in kerosene fuel(Jet-A in this study) proves that the present design of associativeend-groups based on charge-assisted hydrogen bonding (DA/DB and TA/TB inFIG. 22) is successful. FIG. 72, Panel A shows the results of 1 wt %Jet-A solutions of 430 kg/mol NA-, TA- and TB-PCODs, and the 1:1 (w/w)mixture of the 1 wt % solutions of TA- and TB-PCODs at 25° C. It can beseen that self-association of TA remains effective in Jet-A, but it isnot as remarkable as TA/TB association, which gives an increase inspecific viscosity by 270%. These results provide motivation to furtherstudy DA/DB end-association, which is comprised of only 2charge-assisted hydrogen bonds (TA/TB has 4), as an attempt to approachthe limit of the strength of carboxylic acid/tertiary amine association.Fixed at 1 wt % in Jet-A, the results of 200 kg/mol NA-, DA- andDB-PCODs, and the 1:1 (w/w) DA/DB mixture are shown in FIG. 72, Panel B.Comparing the result of the 1:1 DA/DB mixture to that of the control NA,it is found that complementary DA/DB association is also effective inJet-A, and it leads to an increase in specific viscosity by 150%, whichindicates the formation of supramolecules via DA/DB end-association.FIG. 72, Panel C shows that at an Mw of 600 kg/mol, DA/DB associationstill holds, leading to an even higher enhancement of specific viscosity(nearly 200%) relative to the control solution NA. These findings arecontrary to what prior literature teaches us: end-association becomesdifficult when telechelics have long backbones (>100 kg/mol).[63, 64]Taking advantage of the superior strength of charge-assisted hydrogenbonding (˜4 times stronger than ordinary hydrogen bonding),[88] it isable to be realized simple but yet effective pairs of end-groups capableof driving unprecedentedly long chains to form mega-supramolecules inJet-A.

Example 58: 1H NMR Study of Charge Assisted-Hydrogen Bonds

Therefore, charge-assisted hydrogen bonds (CAHB, [73]) that aretypically 3 times stronger than ordinary hydrogen bonds (each CAHBprovides ca. 8-9 kT binding energy) are turned to. Simply placing twotertiary amines at each end of the “di-base” chains (DB) and twocarboxylic acids at each end of the “di-acid” chains (DA) (FIG. 47C)provides an association strength of 16-18 kT ([73]), as recommended bythe theoretical results.

FIG. 65 shows ¹H NMR spectra of isophthalic acid ended (DA) anddi(tertiary amine) ended (DB) polycyclooctadienes (M_(w)=45 kg/mol) and1:1 molar mixture of DA/DB in deuterated chloroform (CDCl₃) indicatingthat carboxylic acid-amine hydrogen bonds dominate over carboxylicacid-carboxylic acid hydrogen bonds. FIG. 65, Panel A, ¹H NMR peaks dueto hydrogens on carbons adjacent to nitrogens of tertiary amine groupsof DB (methyl protons 2; methylene protons 1) shift downfield when theyform charge-assisted hydrogen bonds with carboxylic acid groups of DA(cf. upper to lower spectra: 2 shifts from 2.27 to 2.68 ppm; and 1shifts from 3.59 to 4.13 ppm). FIG. 65, Panel B, ¹H NMR peaks due tohydrogens on the phenyl ring of DA shift upfield upon formation ofcharge-assisted hydrogen bonds between carboxylic acids and tertiaryamines (cf. upper to lower spectra: 1 shifts from 7.96 to 7.84 ppm; and2 shifts from 8.46 to 8.32 ppm). In the present case, the hydrogen ofthe carboxylic acid itself is not observable due to extreme broadeningresulting from rapid exchange with trace H₂O in the solvent. Theformation of acid-amine charge-assisted hydrogen bonds entirely consumesthe available tertiary amine (FIG. 65, Panel A, lower spectrum, nodetectable peak at 3.59 ppm indicates less than 3% of non-associatedamine) and eliminates acid-acid hydrogen bonds (FIG. 65, Panel B, lowerspectrum, no detectable peak at 8.46 ppm indicates less than 3% ofacid-acid association). The absence of acid-acid pairing is consistentwith literature values of the association constants for carboxylic acidself-association (400 M⁻¹) and for charge assisted-hydrogen bonds thatform between tertiary amine and carboxylic acid in chloroform (5×10⁴M⁻¹, [89]).

Example 59: Characterization of Mega-Supramolecule

The formation of mega-supramolecules is evident from solution viscosityand multi-angle laser light scattering (MALLS) measurements. Shearviscosities show that the present longer telechelics do associate intosupramolecules (e.g., at 2 mg/ml in cyclohexane, 300k DA/DB gives ashear viscosity comparable to 670k N_(A), FIG. 48A; this holds fortetralin and Jet-A, as well, FIG. 48B and FIG. 63, Panel A). Even fortelechelics with M_(w) of 670 kg/mol—for which the concentration of endgroups is less than 10 μM (one thousandth of previously studiedconcentrations)([63])—the ends manifestly associate: the viscosity ofthe 670k DA/DB solution is twice that of the non-associative control(FIG. 48A) and multi-million molecular weight supramolecules areconfirmed by MALLS (FIG. 48C and FIG. 63, Panels C-D). At concentrationsas low as 0.22 mg/ml (0.028% wt), 670 kg/mol LTPs form supramoleculeswith an apparent M_(w) of 2,200 kg/mol (FIG. 48C), in accord with themodel prediction that M_(w) corresponds to approximately a three-chainassembly for these conditions, because rings and chains from dimer totetramer dominate (FIG. 64; and for 300k DA/DB, FIG. 63, Panels C-D).Based on the present model, more than ⅓ of the telechelics are inspecies with molecular weight greater than the M_(w) of thesupramolecules. Due to the greater strength of CAHB, acid-base pairingdominates over acid-acid pairing (measured by ¹H-NMR, FIG. 65). Smallangle neutron scattering (SANS) confirms that complementaryend-associative polymers avoid the problem of chain collapse. Theconformation on length scales up to the radius of gyration (R_(g)) ofthe individual chains is just as open for end-associative chains as itis for the corresponding non-associative chains: at q>2π/R_(g)≈0.03 1/Åtheir scattering patterns coincide (FIG. 48D). Together, MALLS, NMR andSANS reveal the molecular basis of the rheological behavior (FIG.48A-B)-complementary end association into mega-supramolecules withexpanded conformations.

Example 60: Phase Behavior of Associative LTPs in Jet-A

Solubility in kerosene over a wide range of operating temperature (−30to +70° C.) is a key requirement for polymers as mist-control additives.One of the major issues with FM-9 polymer contributing to thetermination of the AMK program is that it phase-separates from keroseneeven at ambient temperature, making fuel handling difficult. To test ifthe selection of polymer backbone and end-group structures confers goodlow-temperature solubility in Jet-A, Jet-A solutions of associative LTPsare stored, which are homogeneous at room temperature, in a −30° C.freezer for prolonged periods of time. It is found that even aftermonths of storage at −30° C., all solutions remain homogenous, and nocloudiness due to phase separation of polymer is observed in any sample.Two representative examples are shown in FIG. 15, Panel A and FIG. 73:0.5 wt % Jet-A solution of 264 kg/mol TA-PCOD after storage at −30° C.for 18 months (FIG. 15, Panel A left) and 0.3 wt % Jet-A solution of1:1(w/w) mixture of 430 kg/mol TA- and TB-PCODs (FIG. 73). Clearly theresults suggest the present design of LTPs may overcome the barriers toadopting prior polymers for improving transportation safety andsecurity.

The outstanding solubility of associative LTPs in Jet-A may result fromtwo unique aspects of the molecular design: an unsaturated backbone (seeFIG. 45, Panels A-B and FIG. 47C) and a very low content of polargroups. The multitude of carbon-carbon double bonds in the backboneprovides the host Jet-A with a means to interact with the backbone,leading to the observed good low-temperature solubility without the needof any surfactant or stabilizer. As also shown in FIG. 45, Panels A-Band FIG. 47C, LTPs that show strong end-association in Jet-A have verylittle (≤4) polar groups on each chain end. Take 430 kg/mol TA-PCOD forexample, it contains approximately one oxygen atom per 1,000 carbonatoms. As a result, the occurrence of end-association does not createpolar domains that are large enough to cause phase separation. On thecontrary, FM-9 polymer, which is the mist-control polymer that receivedthe most intensive study to date and has a high content of carboxylicacid group (˜5 mol %) randomly grafted along its backbone, demonstratesa strong tendency to phase separate during storage at ambienttemperature. A package of “carrier fluid” comprised of polar compoundsthat are detrimental to engine operation, including water, glycerol,ethylene glycol, and formic or acetic acid, is needed to keep FM-9barely soluble in Jet-A at ambient temperature.[8, 90] At sub-ambienttemperatures, even the use of carrier fluid cannot prevent FM-9 fromprecipitating from Jet-A. In the context of solution behavior, the sharpcontrast between solution associative LTPs and FM-9 emphasizes the valueof the molecular design shown in FIG. 22 that is based on fundamentalscience.

Example 61: “Shear Degradation” Test and Home-Built Apparatus

Unfortunately, ultra-long backbones undergo chain scission duringroutine handling because hydrodynamic tension builds up along thebackbone to a level that breaks covalent bonds; this “shear degradation”continues until the chains shorten to a point that their valuableeffects are lost (M_(w)<1,000 kg/mol) ([60]). Assembly ofend-associative polymers creates supramolecules that can potentiallybreak and re-associate reversibly, but formation of suchmega-supramolecules (M_(w)≥5,000 kg/mol) at low concentration has neverbeen realized for two reasons: end-to-end association, at lowconcentration, predominantly leads to rings of a small number of chains([83]) and the size of the building blocks is limited because endassociation is disfavored when they are longer than 100 kg/mol([62]-[64]).

In the absence of theory, it was not known whether or not individualchains with lengths below the threshold for shear degradation (1,200kg/mol for PCOD, FIG. 66) and end-association strengths much weaker thana covalent bond (150 kT) could form mega-supramolecules. Theory providea rationale to test telechelics with the predicted end-associationstrength (16-18 kT) and chain lengths, which do form mega-supramoleculeseven at low concentration. They cohere well enough to confer benefitstypically associated with ultra-long polymers-including mist control anddrag reduction. These mega-supramolecules reversibly dissociate underflow conditions that would break covalent bonds, allowing the individualLTPs to survive pumping and filtering, and allowing treated fuel to burncleanly and efficiently in unmodified diesel engines.

FIG. 66, Panel A shows Home-built apparatus for “shear degradation”test. Ultra-long covalent polymers undergo chain scission in intenseflows, such as turbulent pipeline flow and, especially, passage throughpumps. This phenomenon is called “shear degradation.” To subject polymersolutions to conditions that approach the asymptotic limit of sheardegradation (i.e., the backbone length is reduced to the point thatfurther chain scission is very slow), a relatively small volume ofsample (50 ml) is recirculated through a turbine fuel pump at roomtemperature for 60 s (approximately 60 passes through the pump using aflow rate of 3 L/min) using a Bosch 69100 In-line Electric Fuel Pump at12 V. To prevent cross-contamination, the pump was rinsed 4 times withapproximately 200 mL of hexanes, followed by drying under reducedpressure at 40° C. overnight. After recirculation, ‘sheared’ sampleswere collected in 100 mL glass jars and stored at −30° C. FIG. 66, PanelB, An initially 4,200 kg/mol PIB at a concentration of 0.35% in Jet-Ashows the decrease in specific viscosity indicative of shear degradationwith increasing number of passes through the pump. Notice that over 80%of the asymptotic degradation is induced by approximately 60 passes,leading to the selection of the conditions described above. FIG. 66,Panel C, GPC validation of “shear degradation” test using PIB andconfirmation that associative polymers resist degradation.Polyisobutylene having an initial M_(w)=4,200 kg/mol (Before) isdissolved in Jet-A at a concentration of 0.35% wt and recirculatedthrough the fuel pump as described in FIG. 66, Panel A for 60 s(approximately 50 passages through the pump) and the resulting solutionanalyzed by GPC (After). The shift to lower molecular weight(M_(w)=2,300 kg/mol) confirms that the recirculation treatment doesindeed induce shear degradation in accord with the literature onmulti-million molecular weight polymers in dilute solution. The lengthat which the before and after traces cross is the chain length for whichthe rate of degradation matched the rate of production (due to scissionof much longer chains).

A stoichiometric solution of telechelic polycyclooctadienes bearingeither isophthalic acid groups at each end (DA, initial M_(w)=670kg/mol) or two tertiary amine groups at each end (DB, initial M_(w)=630kg/mol) in Jet-A at concentration of 0.3% wt was also analyzed by GPC inas-prepared form (Before; detected M_(w)=747 kg/mol)) and after 60 srecirculation in apparatus (After; detected M_(w)=718 kg/mol). A smalldecrease in the population of the longest chains (fastest elution time;M_(w)≥1,200 kg/mol) may occur. This is considered insignificant as it isnear the detection limit of the instrument; relative to the GPC trace ofthe as prepared DA/DB solution, the GPC trace “after” the recirculationtreatment may also show a minute increase in the population of chains onthe right side of the peak. The latter change is too small to beconfidently measured with the GPC instrument. Note that the possibledegradation of the DA and DB telechelics occurs only where theindividual polymers are so long that they would be vulnerable to sheardegradation. Thus, “stress relief” by reversible dissociation appears toprotect all telechelics <1,200 kg/mol from hydrodynamic chain scission.

Example 62: Shear Stability of LTPs in Jet-A

Fuel is transported through pipes in highly turbulent flow, passesthrough pumps, and needs to be passed through filters in many engines,including aviation turbine engines and large diesel engines. It can becirculated repeatedly through heat exchangers that prevent the enginefrom overheating. In order to ensure that fire protection is retained upto the moment it is needed, degradation prior to fueling or duringfiltering and circulation during operation of the engine can beminimized. Therefore, resistance to flow-induced chain scission (oftencalled “shear degradation”) is among the most crucial requirements formist-control additives for fuels. For linear polymers dissolved in 0-and good solvents, the correlation between shear viscosity and averagemolecular weight of polymer (MW) is well-described by the followingscaling relationship [34]:

η_(s)∝(MW)^(a)

where η_(s) is the shear viscosity and a is the Mark-Houwink constant(0.5 for 0-solvents; 0.76 for good solvents). If a polymer in solutionshear-degrades, such a microscopic phenomenon will be well-reflected bya macroscopic decrease in solution viscosity. Hence, shear viscometryonce again provides a reliable, simple and straightforward method toevaluate shear degradation of polymers in solution after exposure tohigh shear-force environments, such as repeated passage through a fuelpump. Accordingly the setup shown in FIG. 66, Panel A, is used torecirculate the following Jet-A solutions for 60 s (roughly 60 passes)respectively: 4,200 kg/mol polyisobutylene (PIB, a very effectivemist-control polymer but very vulnerable to shear degradation) at 0.35wt %, 430 kg/mol TA-PCOD at 0.3 wt %, and 1:1 mixture of 600 kg/mol DA-and DB-PCODs at 0.3 wt %. Shear viscometry is performed on each solutionbefore and after recirculation, and the results are shown in FIG. 74.

After 60 s of recirculation, the viscosity of the 4,200 kg/mol PIBsolution decreases by 40% (FIG. 74 left; compare the “unsheared” to“sheared”), indicating that the polymer is degraded by shear forceapplied during the test. The results of 4,200 kg/mol PIB not onlyconfirm that PIBs of such a high molecular weight are not shear stable,but also provide a validation that the setup shown in FIG. 66, Panel Acan be used to find out if associative LTPs deliver the promised shearresistance. As shown in FIG. 74 (middle and right), none of the twosolutions of LTPs show detectable decrease in shear viscosity, meaningthat even at an M_(w) of 600 kg/mol, associative LTPs are stillresistant to shear degradation.

The quest for mist-control polymers that survive passage through pumps,filters, and turbulent pipe flow has remained a major unsolved problemdespite decades of research. Typical flow conditions involved in routinefuel handling and transportation are severe enough to degrade ultra-highmolecular weight PIBs and even FM-9, rendering them ineffective.[90, 91]The literature suggests that in dilute (i.e., ˜0.1 wt %) solution, thereis a threshold backbone length (M_(∞)) below which shear degradation ofpolymers does not occur when they are exposed to strong shear, and M.values for polystyrene and PIB are 1,000 kg/mol and 250 kg/mol,respectively. [92-94] For the very reason, even though hard work hasbeen performed to achieve LTPs much longer with respect to priortelechelics, the M_(w) is deliberately kept below 1,000 kg/mol in orderto avoid irreversible chain scission by shear force. Withend-association strengths that are substantially weaker than covalentbonds, supramolecules of LTPs are equipped with “relief valves” thatrespond to turbulent flow by reversibly dissociating, leading to a newclass of potent rheology modifiers that are resistant to sheardegradation.

Example 63: Fuel Treatment with DA/DB for Engine Tests

The current study focuses on mega-supramolecules soluble in low-polarityfluids, especially in liquid fuels. Transportation relies on suchliquids, presenting the risk of explosive combustion in the event ofimpact, such as in the 1977 Tenerife airport disaster—anotherwise-survivable runway collision that claimed 583 lives in thepost-crash fireball. Subsequent tests of ultra-long, associativepolymers (e.g., ICI's “FM-9,” >3,000 kg/mol copolymer, 5 mol % carboxylunits) in fuel increased the drop diameter in post-impact mist ([59],[8]), resulting in a relatively cool, short-lived fire. However, thesepolymers interfered with engine operation ([95]), and their ultra-longbackbone-essential for mist control-degraded upon pumping ([60]).

Unlike ultra-long polyisobutylene (4.2M PIB, 4,200 kg/mol) (FIG. 49A),LTPs survive repeated passage through a fuel pump (FIG. 49B and FIG. 66)and allow fuel to be filtered easily. The acid number, density and flashpoint of the fuel are not affected by mega-supramolecules (FIG. 80).Initial tests in diesel engines indicate that fuel treated with LTPs canbe used without engine modification (FIG. 67): in a long-haul dieselengine (360HP Detroit Diesel), power and efficiency are not measurablyaffected (FIG. 67B). Interestingly LTPs provide a 12% reduction indiesel soot formation (FIG. 49C).

FIG. 67, Panel A shows that the Federal Test Protocol (FTP) for enginetests is a specified transient of RPM and torque designed to includesegments characteristic of two major metropolitan areas in the US. TheFTP cycle consists of four phases (300 seconds each): (1) New YorkNon-Freeway (NYNF, light urban traffic with frequent stops and starts),(2) Los Angeles Non-Freeway (LANF, typical of crowded urban traffic withfew stops), (3) Los Angeles Freeway (LAFY, simulating crowded expresswaytraffic in LA), and (4) a repetition of the first NYNF phase. Initialengine test is performed in double-blind mode, averaging threerepetitions of the FTP cycle with all measurements calibrated betweeneach FTP cycle. The test was performed in diesel engines rather thanaviation jets due to lack of access to an aviation jet engine testfacility. FIG. 67, Panel B, Work and fuel efficiency data using anunmodified long-haul diesel engine at the University of CaliforniaRiverside's Center for Environmental Research and Technology (CE-CERT).Control: untreated diesel. Treated: diesel with 0.14% w/v 670 kg/molDA/DB. BSFC: “brake specific fuel consumption” (fuel burned per workdone against dynamometer, a parameter for fuel efficiency). Bhp-hr:brake-horsepower-hr (0.746 kW-hr). Gal/bhp-hr: gallons per bhp-hr (5.19liters/kW-hr).

Example 64: Long-Haul Engine Test

In the days of the AMK program, all testing aircrafts were required tobe modified with polymer degraders installed before engines because ofthe disastrous effects of FM-9 on engine operation, and this very issueeventually contributed to the termination of the program⁵. The failuresof the AMK program learnt, the significance to have the associative LTPsis fully aware of, no matter they are used as mist-control ordrag-reducing additives, be compatible with unmodified engines. Afull-scale test in a gas-turbine engine would be the ideal way toevaluate the compatibility of LTPs with jet-engine operation; however itrequires approximately 100 barrels of treated jet fuel for eachcomposition and a corresponding total quantity of associative LTPs onthe order of tens of kilograms that is beyond the synthesis capabilityof a university research group. The following facts provide a rationalbasis to use a long-haul diesel engine to test diesel treated with LTPsas a preliminary and affordable means to assess the impacts of LTPs onengine operations: (1) A typical test of fuels in a diesel enginerequires a quantity on the order of 1 barrel. (2) Diesel fuel isconsiderably easier to acquire in large quantity compared to jet fuel.(3) The U.S. Military uses jet fuel to power its fleets of diesel-enginevehicles, which suggests the significance of the interplay between theeffects of LTPs and diesel-engine operation.

Initial tests in diesel engines indicate that diesel fuel treated withthese associative LTPs can be used without any engine modification.Untreated diesel is compared to the same fuel treated with 0.14% w/v 1:1mixture of 600 kg/mol DA- and DB-PCODs using a long-haul diesel engine(360HP Detroit Diesel) and heavy-duty dynamometer (GE 600HP) at theUniversity of California Riverside's Center for Environmental Researchand Technology (CE-CERT). The test is performed in double-blind mode,averaging three repetitions of the Federal Test Protocol cycle with allmeasurements calibrated between each FTP cycle. Power and efficiency arenot measurably affected (FIG. 67, Panel B); the most significant effectof the LTPs is a reduction in production of diesel soot by 12% (FIG.49C). Further testing will be conducted to generate betterunderstandings of the influences of LTPs on the operation, efficiencyand emission of diesel engines powered by diesel, diesel engines poweredby jet fuel, and gas-turbine engines powered by jet fuel.

Measurements were performed at UC Riverside College ofEngineering-Center for Environmental Research and Technology's(CE-CERT's) heavy-duty engine dynamometer laboratory. This enginedynamometer test laboratory is equipped with a 600-hp General ElectricDC electric engine dynamometer. Testing was performed using a DetroitDiesel 360HP engine and the FTP (Federal Test Procedure) heavy-dutytransient cycle for emission testing of heavy-duty on-road engines inthe United States [40 CFR 86.1333]. The FTP transient includes“motoring” segments that take into account a variety of heavy-duty truckand bus driving patterns in American cities, including traffic in andaround the cities on roads and expressways. The FTP cycle consists offour phases (300 s each): (1) New York Non Freeway (NYNF, light urbantraffic with frequent stops and starts), (2) Los Angeles Non Freeway(LANF, typical of crowded urban traffic with few stops), (3) Los AngelesFreeway (LAFY, simulating crowded expressway traffic in LA), and (4) arepetition of the first NYNF phase. The average load factor of the FTPis roughly 20-25% of the maximum engine power available at a givenengine speed. The equivalent average vehicle speed is about 30 km/h andthe equivalent distance traveled is 10.3 km for a running time of 1200s. Fuel was prepared the day before the test. Cans with 3 gallons eachof control and treated concentrates were provided and identified simplyas RED and BLUE to minimize bias during the test and data analysis. Themixture of DA- and DB-PCODs was dissolved at 1.5% in the concentrate.CERT prepared two barrels of identical fuel (25 gal in each barrel). Onthe day before the test, CERT staff added RED can to one barrel and BLUEcan to the other. Mixing was promoted by placing the barrel on a rollerand turning it for approximately 1 hour. The fuel was allowed to standovernight and was used without further mixing during the actual tests.For all tests, standard emissions measurements of non-methanehydrocarbons (NMHC), total hydrocarbons (THC), carbon monoxide (CO),NOx, particulate matter (PM), and carbon dioxide (C02) were performed,along with fuel consumption via carbon balance. The emissionsmeasurements were made using the standard analyzers in CE-CERT'sheavy-duty Mobile Emissions Laboratory (MEL).

Example 65: Impact/Flame Propagation Comparison Tests for TA and PIB

Similarly, high-speed impact experiments (FIG. 69, Panel A) show that,unlike ultralong PIB, LTPs retain their efficacy in mist control afterrepeated passage through a fuel pump. For untreated Jet-A fuel, theimpact conditions generate a fine mist through which flames rapidlypropagate into a hot fireball within 60 ms. Polymer-treated fuel samplesare tested in two forms: as prepared (“unsheared”) and afterapproximately 60 passes through a fuel pump (“sheared”) (FIG. 66).Ultra-long PIB (4,200 kg/mol, 0.35% wt) is known to confer mist controlthat prevents flame propagation (FIG. 50A, top left; [7]); however,“sheared” PIB loses efficacy (FIG. 50A, top right). LTPs (TA, propertiesshown in FIG. 70, Panel A) provide mist control both before and aftersevere shearing (FIG. 50A bottom), confirming their resistance to sheardegradation (FIG. 70, Panel B). The qualitative effects seen in stillimages at 60 ms (FIG. 50) are quantified by computing the averagebrightness of each frame (3,000 images in 300 ms), which shows that both“unsheared” and “sheared” TA-treated fuels control misting (FIG. 69,Panel C). Moreover, the test also proves that chain length of thetelechelics plays a crucial role in mist control (FIG. 50B), consistentwith the hypothesis that mega-supramolecules are the active speciesconferring the observed effect.

FIG. 69, Panel A shows apparatus for impact/flame propagationexperiments. An aluminum canister (outer diameter=23 mm, height=100 mm)was used as a miniature fuel tank to hold ˜30 mL of a test sample. Thecap was tightly sealed with superglue and electrical tape. A stainlesssteel cylinder (diameter=24 mm, length=50 mm) was used as a projectileto impact the sample canister and disperse the fuel. To the left of thisimage: Compressed air at 6.89×10⁵ Pa was used to propel the projectilethrough a 1.66 m-long barrel (inner diameter=25.4 mm), resulting in amuzzle speed of 63 m/s measured by time of flight between twoflush-mounted sensors in the barrel. An array of three continuouslyburning propane torches was placed in the path of the ejected fuel. Toprevent the torches from being extinguished by the burst of air from thegun, a shield was placed between the torch tip and the gun. The impact,misting, subsequent ignition and flame propagation were captured using ahigh-speed camera (Photron SA1.1, frame rate 10 kHz). Image acquisitionwas triggered by a laser-motion detector attached to the end of muzzle.

FIG. 69, Panel B shows frame at 60.4 ms for untreated Jet-A. Therectangular box is the area within which pixels were analyzed forbrightness.

FIG. 69, Panel C shows average brightness of the pixels in the rectanglebox of FIG. 69, Panel B as a function of time during the first 300 msafter impact for five compositions (untreated Jet-A, 0.35% wt 4.2M PIBunsheared, 0.35% wt 4.2M PIB sheared, 0.3% wt 430 kg/mol TA unshearedand 0.3% wt 430k TA sheared). The brightness of each pixel was scaledfrom 0 to 250. The average brightness of the pixels in the rectangularbox (shown in part FIG. 69, Panel B) was calculated for each frame(every 0.1 ms). Untreated Jet-A generated a large fireball (almost allpixels in the red rectangle were saturated) that was relatively longlasting (intense flame from 40 ms to 60 ms, followed by a prolonged timein which separated flames continued to burn until all fuel wasconsumed). As-prepared 4.2M PIB suppressed flame propagation, but lostits efficacy after the shear treatment described in FIG. 66. 430 kg/molTA was effective in mist-control before and after shear.

Example 66: Impact/Flame Propagation Test

Associative LTPs are proven to be highly effective in mist control,preventing flame propagation in post-impact jet fuel mist. The apparatusshown in FIG. 69, Panel A is used to emulate the impact-inducedatomization and subsequent ignition of kerosene released from rupturedfuel tanks in crash scenarios of ground vehicles/aircraft. A steelprojectile is shot at 63 m/s at a sealed aluminum tube containing thefuel sample to generate mist, while three propane torches are burningalong the path of the ejected fluid. The process of impact, misting,ignition and flame propagation is captured using high-speed imaging.

Efficacy of high molecular-weight end-associative polymers asmist-control additives for fuels was studied via high-speed imagingduring an impact/flame progagation test. The apparatus (FIG. 69, PanelA) emulates the atomization and subsequent ignition of fuels releasedfrom ruptured fuel tanks in crash scenarios of ground vehicles/aircraft.An aluminum canister (outer diameter=23 mm, height=140 mm) pre-loadedwith a cylindrical aluminum filler (diameter=22 mm, height=40 mm) wasused as a miniature fuel tank to hold ˜30 mL of a test sample. The capwas tightly sealed with superglue and 2-3 wraps of electrical tape tokeep it in place during the impact. A solid stainless steel cylinder(diameter=24 mm, length=50 mm) was used as a projectile to impact thecanister and disperse the fuel. Compressed air at 6.89×105 Pa was usedto propel the projectile through a 1.66 m-long barrel (innerdiameter=25.4 mm), resulting in a muzzle speed of 63 m/s. An array ofthree continuously burning propane torches was placed in the path of theejected fuel to serve as ignition sources. The onset of impact,formation of mist, and the following ignition events and propagation offlame were captured at a frame rate of 10 kHz using a high-speed camera(Photron SA1.1). Image acquisition was triggered by a laser-motiondetector attached to the end of muzzle.

For untreated Jet-A, the impact conditions generate a fine mist: at 30ms after the impact, a cloud of very fine mist of Jet-A is observed(FIG. 75, Panel A), and at 60 ms after impact flames rapidly propagatethrough the fine mist into a hot fireball (FIG. 75, Panel B). The flamepropagated to engulf the entire cloud of fuel mist within a further 60ms. Polymer-treated Jet-A samples are tested in two forms: as prepared(“unsheared”) and after being passed through a fuel pump approximately60 times (“sheared”) using the setup shown in FIG. 66, Panel A. Theultra-long 4,200 kg/mol PIB at 0.35 wt % in Jet-A is used as a positivecontrol that is known to confer mist control which prevents flamepropagation. As shown in FIG. 76 (left), much larger dropletsinterconnected by fluid filaments are observed at 30 and 60 ms afterimpact. As ejected fluid flies over the propane torches, localizedignition events are observed, but they soon self-extinguish. The“sheared” sample of the 0.35 wt % Jet-A solution of 4,200 kg/mol PIB,however, shows a significantly different pattern of ejection of fluidafter impact (FIG. 76 right): fine droplets formed and interconnectingfilaments are no longer observed. Ignition events observed at 30 msafter impact quickly propagate and engulf the fuel cloud in fireball(FIG. 76 right, t=60 ms), indicating the polymer loses its efficacy dueto shear degradation. The results confirm that the method is capable ofcreating a post-impact fuel mist that propagates fire from any ignitionevent, correctly captures the fire protection that is known to beconferred by 4,200 kg/mol PIB at 0.35 wt % and the loss of fireprotection that is known to occur after fuel is passed through pumps,filters or turbulent pipe flow.

Having validated the setup, it is used to test the efficacy ofassociative LTPs as mist-control additives for Jet-A. Here 430 kg/molTA-PCOD is selected as a representative example. The results in FIG. 77prove that associative LTPs provide mist control both before and aftersevere shearing, confirming their resistance to shear degradation. It isfound that in the test of the unsheared solution of 430 kg/mol TA-PCOD,supramolecules suppress mist formation of Jet-A: ignition eventsself-extinguish and, as a result, no propagating fireballs are observed.When the sheared solution is tested, the post-impact ignition eventspropagate to a very limited extent (FIG. 77 right, t=60 ms), and they donot evolve into a propagating fireball at all, evidencing that themist-control ability of the polymer remains after going through severeshearing. Moreover, the test also proves that chain length ofassociative LTPs plays a crucial role in mist control, consistent withthe hypothesis that mega-supramolecules are the active speciesconferring the observed effect. Unsheared 0.5 wt % Jet-A solutions ofTA-PCODs at Mw=76, 230, 300 and 430 kg/mol are tested, and completesuppression of fire propagation is only observed in the case of thelongest TA polymer (FIG. 50B).

The results shown in FIG. 77 clearly indicate that associative LTPs doavoid the problem of chain collapse resulting from randomly placingassociative groups along polymer backbone.[74, 96] If not, propagatingfireballs would have been observed in tests of 430 kg/mol TA-PCODsolutions. In accord with theoretical predictions that very longbackbones reduce cyclic association and favor intermolecular associationeven at low concentration, the results show that increasing the lengthof long telechelic associative polymers favors formation elasticsupramolecules at low concentrations and confers mist control.[97]Hence, overcoming synthetic obstacles to long (>300 kg/mol) telechelicassociative polymers is proved to be significant, for it provides accessto the unexplored regime of very long LTPs (>400 kg/mol) that cancontrol misting of kerosene like ultra-high molecular weight PIBs andsurvive turbulent flows that can destroy ultra-high molecular weightPIBs.

In the 70's and 80's, the prevailing concept for improving fire safetyof fuels was that it could be achieved through the addition ofthen-known anti-misting polymers into fuels to completely eliminate theimpact-induced atomization of fuels and the subsequent fire/explosionhazards.[7, 8, 90, 98, 99] However, more recent studies indicate thatsimply shifting the drop size distribution to higher values can preventflame propagation through a fuel mist. For example, the critical valuesof Sauter mean diameter of droplets of military fuel JP-8 in adroplet/air (aerosol) mixture to propagate a flame from an ignitionsource is approximately 52 m; at lower droplet sizes than this criticalvalues the aerosol becomes entirely engulfed in flame.[100] Thus,complete elimination of mist formation is not necessary. This idea is ingood agreement with the data shown in FIGS. 75-77. Even though fireresistance are observed in both unsheared solutions of 4,200 kg/mol PIBand 430 kg/mol TA-PCOD, impact on the latter results in a cloud of finerdroplets compared to the former. The observed fire protection conferredby 430 kg/mol TA-PCOD clearly indicates that it does not requirecomplete elimination of misting to achieve fire-safe fuels; instead, thegoal can be achieved via proper control of misting.

Motivated by the hope to prevent the use of civilian aircrafts asweapons of mass destruction, long telechelic polymers (LTPs) wereexplored and it was demonstrated that their length is key to LTPs'potent rheological effects. It is found that by carefully selectingassociative end-groups that associate with a strength much greater thanthermal energy (kT), yet much weaker than a covalent bond (ca. 150 kT),LTPs form mega-supramolecules even at low concentration. Thesesupramolecules provide benefits typically associated with ultra-longpolymers-including mist control and drag reduction, and they reversiblydissociate under flow conditions that would break covalent bonds,allowing the individual LTPs to survive pumping and filtering andallowing treated fuel to burn cleanly and efficiently in unmodifieddiesel engines. After a 30-year gap in polymer research to improve firesafety and stewardship of fuel, LTPs represent an “existence proof” thatpolymers can indeed control misting and reduce pumping costs withoutlosing efficacy due to shear degradation, or harming fuel economy oremissions.

Example 67: Effect of TA on Specific Viscosity of Tetralin

FIG. 70 shows characterization of α,ω-di(di(isophthalic acid)) (TA)polycyclooctadiene used in Impact test. FIG. 70, Panel A, Effect ofchain length on specific viscosity of TA in tetralin at 10 mg/ml. FIG.70, Panel B, Specific viscosity of 2.4 mg/ml 430 kg/mol TA in Jet-A at25° C., sheared vs unsheared. The 430 kg/mol α,ω-di(di(isophthalicacid)) polycyclooctadiene (TA), which is used in the impact test, isself-associative (and might not be pairwise). Although its physics maydiffer from that of complementary pairs, its rheological properties aresimilar FIG. 70, Panel A, it has similar resistance to shear degradationFIG. 70, Panel B, as the α,ω-di(isophthalic acid) polycyclooctadiene andα,ω-di(di(tertiary amine)) polycyclooctadiene 1:1 molar ratio mixture(˜670 kg/mol DA/DB).

Example 68: Safer and Cleaner Fuel by End-Association of Long TelechelicPolymers

Liquid fuels, such as gasoline, diesel and kerosene, are the world'sdominant power source, representing 34% of global energy consumption.Transportation relies on such liquids, presenting the risk of explosivecombustion in the event of impact, such as the 1977 Tenerife airportdisaster—an otherwise-survivable runway collision that claimed 583 livesin the post-crash fireball. The UK and the U.S. responded with amulti-agency effort to develop polymeric fuel additives for “mistcontrol.” Ultra-long, associative polymers (e.g., ICI's “FM-9,” >3,000kg/mol copolymer, 5 mol % carboxylic acid units) increased the dropdiameter in post-impact mist, resulting in a relatively cool,short-lived mist fire. However, the polymers interfered with engineoperation, and their ultra-long backbone-essential for mistcontrol-degraded upon pumping. They were abandoned in 1986. 15 yearslater, the post-impact fuel fireball involved in the collapse of theWorld Trade Center provided motivations to revisit polymers for mistcontrol.

Building on recent advances in supramolecular assembly as a route toemergent functional materials, particularly assembly of complex polymerarchitectures, an unexplored class of polymers that is both effectiveand compatible with fuel systems was discovered. Here, it is shown thatlong (>400 kg/mol) end-associative polymers form “mega-supramolecules”that control post-impact mist without adversely affecting power,efficiency or emissions of unmodified diesel engines. They also reduceturbulent drag, hence, conserving energy used to distribute fuel. Thelength and end-association strength of the present polymers weredesigned using statistical mechanical considerations. In comparison withultra-long polymers for mist control, the present polymers are an orderof magnitude shorter; therefore, they are able to resist sheardegradation. In contrast to prior randomly-functionalized associativepolymers, these end-associative polymers also avoid chain collapse. Itis found that simple carboxylic-acid/tertiary-amine end-association iseffective, and the unprecedented length of these telechelic polymers isessential for their potent rheological effects.

Kerosene fuels have been a major source of fire hazard and vulnerabilitywhen they are released in an uncontrolled manner. It is estimated that40% of the fatalities in so-called “survivable aircraft crashes,” whichmake up approximately 70% of accidents that occur on takeoff andlanding, are due to fire caused by combustion of aviation fuel.[101]Similarly, the violent and catastrophic combustion of leaked fuel afterthe direct or indirect ballistic penetration of a vehicle's fuel tank orfuel line by shrapnel in IED attacks has inflicted heavy casualties onUS military over the last decade. In impact scenarios, fuel is atomizedby mechanical energy involved into fine mist, and such mist burnsexplosively when ignited. The resultant fire can rapidly propagate awayfrom the ignition source, involve more fuel, and trigger deadly poolfires that are very violent and difficult to contain. Such fire oftenaccompanies tank explosions, leaving no chance for firefighters tointervene, as demonstrated in the collapse of the World TradeCenter.[102]

Increasing the droplet size in post-impact mist of kerosene (i.e., “mistcontrol”) has been identified as the most promising way to mitigateimpact-induced kerosene fires.[103],[104] “Mist-control kerosene” isindeed a fuel that “burns but doesn't burn—” after ignition from anincendiary threat, it self-extinguishes and slows the spread of fire sothat fire-extinguishing systems can intervene, and personnel can havetime to escape.[91] Ultra-high molecular weight (on the order 10,000kg/mol) polymers have potent effects on the breakup of liquid jets anddrops even at very low concentration (on the order of 100 ppm),[105]since they are long enough to exhibit elasticity and sustain tensilestress.[15, 106-108] However, using such polymers to provide mistcontrol for kerosene has been found practically difficult due to theirvulnerability to shear degradation in fuel transportation and dispensingprocesses.[109] Once they are degraded, they lose their efficacypermanently. Beginning in the late 1970's, efforts had been made toadopt the concept of “associative polymers” in hope of providingmist-control effectiveness of ultra-high molecular weight polymers whilecircumventing their loss of efficacy due to shear degradation.Specifically, these associative polymers are comprised of shear-stablepolymer chains (molecular weight≤1,000 kg/mol) with associative groupsrandomly placed on the backbone, capable of aggregating into largerclusters (which might be effective in mist control) via hydrogen bondingand responding to turbulent flow via reversible dissociation.[110] Agood example is ICI's FM-9 polymer (>3,000 kg/mol copolymer, 5 mol %carboxylic acid units) used in the engineering-oriented UK-U.S. jointAnti-Misting Kerosene (AMK) program.[90, 91] Despite demonstration ofefficacy, FM-9 interfered with engine operation and fuel handling, andit was not immune to shear degradation upon pumping. Eventually researchin this area was largely abandoned in 1986.

With a view towards improving fire safety of jet fuels, polymers formist control of kerosene are disclosed herein. Fundamental relationshipbetween molecular designs of mist-control polymers and correspondingsolution behaviors are described. It was found that the design ofassociative polymers prevailing in the 80's suffers a fatal flaw: itleads to self-assembly of chains into collapsed, inextensible structuresthat are of little use in mist control.[74, 96] Using statisticalmechanical analysis of ring-chain equilibrium, it is found thatexceptionally long telechelic polymers (LTPs, FIG. 22) are needed toassemble large supramolecules effective in mist control and otherapplications that rely on large polymer coils, such as dragreduction.[97] The theory provides a guideline on polymer backbonelength based on two trade-offs: the chain is typically be long enoughthat mega-supramolecules form, yet short enough to avoid chain scission(<1,000 kg/mol) during pumping and turbulent flow. Specifically, it isexpected that an adequate concentration (>50 ppm) of >5,000 kg/molsupramolecules forms when the individual telechelics are 500 kg/mol, ifdonor-acceptor type associations are used with end-association energyapproximately 16-18 kT, and the total polymer concentration is 1,400ppm. These criteria point to an unprecedented class of polymers that didnot exist before.

In the present disclosure, the recent advance in development of LTPs formist control of kerosene are described, including the breakthrough inpolymer and supramolecular chemistry and the properties and performanceof these polymers as fuel additives.

Example 69: Identification of Associative Polymers to Control DragReduction in Aviation Fuel

The following approach can be used to identify associative polymers fordrag reduction in aviation fuel, in particular in to achieve a 10%increase in pipeline capacity through the existing pipelines serving anairport.

Candidate polymer backbones can be identified for a certain fuelcomposition to be used as a host composition in the sense of the presentdisclosure. For example a skilled person can refer to literature toidentify polyisobutylene (PIB) as a candidate known to be widely used infuel additives and be able remains in solution down to low temperatures(e.g. −30° C.). Additional candidates (e.g. polycyclooctadiene) can befound based on literature or experiments to be performed to identify thesolubility and stability of the polymer in a fuel composition of choice.

Prioritization (rank order) of the candidate backbones can be achievedby the following steps

-   -   Step 1. For each candidate backbone, identify the threshold        molecular weight for the onset of shear degradation. This        provides a good estimate for the maximum span of associative        polymers herein described, whether linear or branched, suitable        for their application.    -   Step 2. For each candidate backbone, determine (e.g. by        measuring) the overlap concentration that corresponds to the        maximum span of the associative polymers suitable for their        application, determined in Step 1. If a backbone shorter than        the maximum is used, it will increase the value of the overlap        concentration. So the overlap concentration determined in this        step is the lowest overlap concentration relevant to their        application.    -   Step 3. For each candidate polymer backbone, determine the end        group concentration that corresponds to the overlap        concentration determined in Step 2.    -   Step 4. For each candidate polymer backbone, estimate the range        of the association constants worthy of testing. Specifically,        using the molar concentration of end groups from Step 3,        determine the value of the association constant that would        provide a pairing of the end groups equal to or greater than 75%        (e.g. 99%) according to the binding constant calculated in        accordance with the present disclosure. If the polymer will be        tested at concentrations below c*, the association constant        estimated using c* provides a lower estimate of the association        constant that will give the desired effects. If polymers will be        used at higher concentration than c*, the association constant        estimated using c* will provide desired effects.

Results of application of the above approach to the exemplary PIB andPCOD indicated above is summarized in the following Table 15 below

Results of application of the above approach to the exemplary PIB andPCOD indicated above is summarized in the following Table 15 below

TABLE 15 Threshold Overlap cnc. Association constant Candidate M_(w)[g/mol] [g/L] at End group range of interest backbone for shear R_(g)[nm] for threshold conc. [M] For 75% For 99% structure degradationthreshold Mw Mw & R_(g) at c* end assn. end assn. PIB 300,000 25 7.7 2.6× 10⁻⁵ 2 × 10⁵ 1.2 × 10⁸ PCOD 700,000 73 2 2.3 × 10⁻⁶ 5 × 10⁶   3 × 10⁹

A skilled person can also perform experiments to identify the thresholdmolecular weight for their application. For example, an apparatus can beconstructed that subjects the fluid to the number of passes through apump, the exposure to turbulent pipe flow and passage through filtersthat is pertinent to the application of interest to them. Alternatively,the skilled person can perform a literature search to obtain an estimateof the value of the threshold molecular weight for each backbone ofinterest. For illustration, exemplary estimates for PIB and PCODobtained from laboratory experiments are provided in the second columnof Table 15 above.

The threshold molecular weight if the architecture are linear, given inthe second column of table 15 (the longest span—see e.g. FIGS.81A-81H—for the application of interest), can be used to determine (e.g.by calculation or measurement) the corresponding radius of gyration,shown in the third column of the table. The radius of gyration R_(g)calculated for a linear chain corresponding to the longest span providesa good estimate of the radius of gyration for the other polymerarchitectures of the present disclosure. The skilled person can eitherperform experiments to measure R_(g) for the backbones of interest andobtain the value of R_(g) that corresponds to the threshold molecularweight in the second column of table 15. Alternatively, the skilledperson can refer to the literature and their knowledge of the solutioncondition relevant to the candidate backbones.

In the present example, fuel is a good solvent for both of the backbonesbeing considered. The values shown in the third column of Table 15 werecalculated for good solvent conditions using equations provided forpolybutadiene and polyisobutylene as equations (6) and (26) in“Molecular Weight Dependence of Hydrodynamic and ThermodynamicProperties for Well-Defined Linear Polymers in Solution” (1994) byFetters et al. [23]

The threshold molecular weight and the corresponding radius of gyrationcan be used to calculate the minimum overlap concentration that can beachieved with each candidate backbone, limited by their individualthreshold for shear degradation under the condition of the user'sapplication. As noted above, the R_(g) calculated from the longest spanprovides a good estimate of the radius of gyration for the other polymerarchitectures of the present disclosure. In the exemplary case of PIBand PCOD, the M_(w) used to calculate the concentration in the fourthcolumn of the table assumes that the polymers are linear. A skilledperson would know how to determine M_(w) for other architectures fromthe size of the longest span and the specific architecture of interest.

The end group concentration for the threshold molecular weight at theoverlap concentration can be determined (e.g. by calculation ormeasurement). In this example, the case of a linear associative moleculeis used and complementary association (A+B pairwise association) isassumed: each polymer has two ends; half of the polymers carry the Afunctional group and half carry the B functional group. Thus, the molarconcentration of A ends equals the molar concentration of B ends equalsthe molar concentration of chains, given in the fifth column of thetable. The skilled person can adjust this as appropriate to theassociative molecules of interest to them, which might have more thantwo functional groups if branched structures are considered (see e.g.FIGS. 81A-81H) and might be self-associative or involve more than twocomplementary functional groups.

In the example given in table 15, the relevant range of associationconstants is calculated assuming pairwise, complementary association, asdescribed in the preceding paragraph. Thus, the values given in thesixth and seventh columns of the table are equal to (0.75 [end])/{(0.25[end])²} for the 75% case and (0.99 [end])/{(0.01 [end])²} for the 99%case, where [end] denotes the end group concentration value given in thefifth column. The skilled person would be able to adjust the calculationas appropriate to other scenarios, also described above.

A skilled person can now prioritize the experiments to be performed todevelop the formulation that meets the required 10% reduction inpipeline drag. For example, if the concentration needs to be kept below3 g/L, then the skilled person may exclude PIB from furtherconsideration. Initial experiments may focus on linear PCOD with Mw andPDI such that less than 1% of chains are longer than 700 kg/mol.Experiments can focus on end group structures that give associationconstant greater than 4.9×10⁶. The reduction of pipeline drag can thenbe measured for a small number of concentrations, perhaps c*, c*/2 andc*/4, to characterize trends in performance as a function ofconcentration. If the effects are not adequate, a stronger associationconstant can be tested. If the resistance to shear degradation is notadequate, a branched architecture can be tested. The skilled person canuse a relatively modest number of experiments to develop a polymer andformulation that meets the requirement for 10% reduction in pipelinedrag.

Example 70: Associative Polymers to Control Droplet Breakout DuringFibers' Preparation

A skilled person seeks to prepare fibers using electrospinning of anonpolar monomer, ethylhexylmethacrylate. The liquid undergoeselectrospray into fine droplets rather than electrospinning. The skilledperson adds 0.1% of 700k DA/DB to the monomer. The problem of dropletbreakup is eliminated, enabling spinning of the desired fiber. When thefiber diameter is drawn down to 80 nm, photopolymerization is used tosolidify the fiber.

Example 71: Associative Polymers to Control Size and Uniformity of DrugParticles

A pharmaceutical company uses atomization of hydrophobic drug in anon-polar solvent followed by evaporation of the nonpolar solvent toproduce particles of the drug. The size and uniformity of the drugparticles can be used to optimize their time release when administeredto the patient. A skilled person seeks to eliminate satellite droplets.The skilled person chooses as the backbone of the associative polymerherein described a hydrophobic polymer accepted for use in drugformulations and soluble in the drug solution used for atomization. Theskilled person identifies 10 g/L concentration as the acceptable amountof polymer in the drug solution used for atomization. Therefore, theychoose a polymer molecular weight that gives the polymer a radius ofgyration of 22 nm. They consider functional groups in relation to thecomposition of the atomization solution to select functional groups thatwill associate with association constant k>10⁵ when used in thatsolution. The polymer is introduced to the solution at a concentrationof 10 g/L and the formation of satellite drops is reduced.

Example 72: Associative Polymers to Increase Volume of a Fluid Suppliedin a Pipeline

A fuel pipeline is operating at its maximum capacity. A skilled personwants to increase the volume of fuel supplied through the pipeline. Thepipeline is operating at its maximum pressure, so the increase inthroughput cannot be accomplished by increasing the pressure. The flowin the pipeline is turbulent (the Reynolds number is greater than 5,000,e.g. 25,000). Therefore, frictional losses in the pipeline are describedusing the familiar friction coefficient C_(f). defined as

$\begin{matrix}{C_{f} = {\frac{{Wall}\mspace{14mu}{Shear}\mspace{14mu}{Stress}}{{Dynamic}\mspace{14mu}{Pressure}} = \frac{2{D\Delta p}}{4{L\rho u}_{m}^{2}}}} & (26)\end{matrix}$

where D is the inner diameter of the pipe, Δp/L is the frictionalpressure loss over a distance L along the pipeline, ρ is the density ofthe fuel, u_m=Q/A, where Q is the volumetric flow rate and A=πD²/4 isthe cross sectional area of the pipe. Often the frictional pressure lossis expressed as “head loss” h_(f)=Δp/(ρg):

$\begin{matrix}{{h_{f} = {\frac{4C_{f}{Lu}_{m}^{2}}{2{gD}} = {\frac{4C_{f}{LQ}^{2}}{2{gDA}^{2}} = {\frac{32C_{f}{LQ}^{2}}{{g}^{2}D^{5}} = {RQ}^{2}}}}}\mspace{14mu}{R\mspace{14mu}{is}\mspace{14mu}{the}\mspace{14mu}{fluid}\mspace{14mu}{resistance}}} & (27)\end{matrix}$

Laboratory experiments were performed at a Reynold's number of Re=14,000using 9 m long and 12 m long tubes. Compared to untreated fuel, thefluid resistance due to flow through the tube was reduced fromhf/Q²=1.1×10¹¹ s²/m⁵ (untreated fuel) to hf/Q²=6.8×10¹⁰ s²/m⁵5 whentreated with 0.1% of a 1:1 mixture of 700k DA and 700k DB.

In the pipeline, the Reynold's number is much greater, approximatelyapproximately 100,000. In accord with prior literature indicating thatthe fractional drag reduction increases with Re over this range, whenthe polymer was used in the pipeline, the increase in throughput wasmore than 25%.

Example 73: Associative Polymers to Provide Grafting Sites on a FiberSurface

In an exemplary application, a hydrophobic polydrug is only available inmolecular weights that are too short to enable fiber spinning. Inaddition, a covalently grafted layer is needed on the surface to inhibitnon-specific protein adsorption. A product development team seeks asingle additive that can be used at low concentration to providegrafting sites on the fiber surface. Therefore, the team chooses abranched polymer with the following average structure:

On average the polymers have four nodes. On average they have fourassociative functional groups, FGas. In addition, on average, eachmolecule has one FGd. On average they have nine -[chain]- segments eachapproximately 100 kg/mol, such that the average molecular weight of thepolymer is approximately 1,000 kg/mol. When 0.3% of the above is addedto the solution, the associative polymer facilitates fiber spinning andprovides FGd groups at the fiber surface. The FGd groups displayed onthe surface of the fiber are later used as chemical groups for graftingPEG or zwitterionic polymer chains to the fiber surface.

Example 74; Determination of F_(b) Using Cross Slot

The rupture force for polystyrene is measured using a cross slot flowdevice of the design described by L. Xue, U.S. Agarwal, P. J. Lemstra,“Shear Degradation Resistance of Star Polymers during ElongationalFlow,” Macromolecules, 38, 8825-8832 (2005) as shown in FIGS. 91 A andB. The starting material has Mw=8470 kg/mol and Mn=3940 kg/mol asmeasured by gel permeation chromatography (FIG. 92, Panels A-C, solidcurve). A solution of the polystyrene sample is prepared in decalin viamagnetic stirring under argon atmosphere at a concentration of 100 ppm(w/v). 2000 ml of the solution is placed into the high-pressurereservoir in FIG. 91C. A nitrogen cylinder with a pressure regulatorappropriate for 1-5 bar is connected to the reservoir. The experiment isrun by opening the valve to apply pressure to the fluid in thereservoir, which drives the solution through two opposing slots into thecross-slot flow cell, and then out through the other two opposing slots,and into the collection reservoir. The “flow time” required to drive 2Lof sample through the system is measured. Adjust the pressure so theflow time for decalin alone is approximately 10 s (the highest flow rateof the series of experiments)

Load the reservoir with 2L of the polystyrene solution. Open the valveand measure the time required to move 2L from the high-pressurereservoir to the collection reservoir. At the end of each pass, collect20 ml sample from the test solution in the collection reservoir. Put theremaining solution in the reservoir and repeat.

Note the decrease of the flow time with successive passes. When the flowtime no longer changes, the series of experiments is complete. Discardthe spent solution.

Place 2L of as-prepared solution in the pressure reservoir. Adjust thepressure to a value that is ⅔ of the pressure applied during the firstseries of experiments. Open the valve and measure the time required tomove 2L from the high-pressure reservoir to the collection reservoir. Atthe end of each pass, collect 20 ml sample from the test solution in thecollection reservoir. Put the remaining solution in the reservoir andrepeat. Note the decrease in flow time with successive passes; a largernumber of passes is required for the flow time to stop changing. Once itstops changing discard the spent solution.

Place 2L of as-prepared solution in the pressure reservoir. Adjust thepressure to a value that is ⅔ of the pressure applied during the secondseries of experiments. Open the valve and measure the time required tomove 2L from the high-pressure reservoir to the collection reservoir. Atthe end of each pass, collect 20 ml sample from the test solution in thecollection reservoir. Put the remaining solution in the reservoir andrepeat. Note the decrease in flow time with successive passes; an evenlarger number of passes is required for the flow time to stop changing.Once it stops changing discard the spent solution.

Aliquots are selected for analysis based on the number of passes thatwere required for the flow time to stop changing. If fewer than 20passes were required, analyze each of the first 6 aliqots, the lastaliquot and one half the number of passes between the 6^(th) and thelast pass. If approximately 100 passes were required, analyze aliquotsaccording to the geometric series: #2, #4, #8, #16 etc.

Polymer in the aliquots selected for analysis is recovered by adding 30ml of methanol into the aliquot, followed by centrifuging the resultingmixture at 2,500 rpm for 10 min and discarding the supernatant.Subsequently, the recovered polymers are dissolved in tetrahydrofuran(THF) at a concentration 1 mg/ml, and the resulting samples arecharacterized using a gel-permeation chromatography (GPC) instrumentthat is equipped with a multi-angel laser light scattering (MALLS)detector. Representative GPC traces are shown in FIG. 92. The resultsprovide the average molecular weight and molecular-weight distributionof the recovered polymers.

The asymptotic values of the degraded Mw are used to compute the forcerequired to break the polystyrene backbone as follows.

The density 824 kg/m³ and viscosity 1.3×10⁻³ Pa·s of decalin at thetemperature of the test (25 C) are used to evaluate the Reynolds number.

For calculation of the Reynolds number, the velocity of the flow iscalculated using the volumetric flow rate of the last run in eachseries:

U=Q/(d·l), using the gap d=0.3 mm and the depth l=2.5 mm of the channelsin the apparatus.

The volumetric flow rates of the last run in each of the three seriesare: 150 ml/s, 21.8 ml/s, 13.2 ml/s.

The resulting values of the Reynolds number for the three runs are:2.94×10⁴, 4.29×10³, and 2.59×10³.

The asymptotic values of M_(w) are: 153 kg/mol, 830 kg/mol and 1200kg/mol. The corresponding contour lengths are calculated by dividingM_(w) by the monomer molecular weight M_(o)=108 g/mol, which yields thenumber of monomers in the corresponding chain. The number of monomers isconverted to the number of backbone bonds by multiplying by n_(o)=2(each styrene repeat unit contributes two backbone carbons). The numberof backbone bonds is converted to contour length by multiplying by 0.126nm (the product of the length of a C—C single bond and sin(109°/2) forsp³ carbon):

Polystyrene contour length values corresponding to the observedasymptotic M_(w) are: L=358 nm, 1940 nm, 2810 nm.

This provides all of the quantities required to evaluate thehydrodynamic force that was acting on chains of the asymptotical M_(w)for each flow condition:

$F_{K} = \frac{{\mu}^{2}{Re}^{3/2}L^{2}}{4\rho\; d^{2}{\ln\left( {{L/1}\mspace{14mu}{nm}} \right)}}$

Three experimentally determined values of F_(K) are calculated using thecontour length and Reynolds number values for each run. The resultingvalues of FK are: 3.55 nN, 4.40 nN, 4.12 nN.

The average of the three values provides a suitable value for thebreaking force of a polystyrene backbone, F_(b): Polystyrene F_(b)=4.02nN

This value can now be used to design framing and capping chains based onpolystyrene as illustrated in subsequent example(s).

Example 75: Hydrodinamic Forces in a Flow and Related Calculation

The hydrodynamic force exerted on a polymer, particularly at aconcentration less than c* in a nonpolar composition in a flow, dependson the viscosity host non-polar composition. A table showing viscosityvalues for exemplary host composition is shown in FIG. 82. Inparticular, the viscosity of the host non-polar composition has aproportional effect on the hydrodynamic force for a given deformationrate.

If an associative polymer herein described having a contour length ofits longest span equal to 730 nm passes through a turbulent eddy wherethe elongation rate is 3200 s⁻¹, in a nearly fully extended conformationthe polymer would be subjected to a tension force F_(K)≈(120 nm)² (3200s⁻¹) (0.650 N·s/m²) (10⁻⁹ m/nm)=1.1 nN if the host non-polar compositionis Castor Oil.

The same polymer (contour length of its longest span equal to 730 nm)passing through a turbulent eddy where the elongation rate is 3200 s⁻¹,in a nearly fully extended conformation the polymer would be subjectedto a much lower tension force F_(K)≈(120 nm)² (3200 s⁻¹) (0.00164N-s/m²) (10⁻⁹ m/nm)=0.0028 nN if the host non-polar composition iskerosene.

Example 76: Density and Viscosity of a Non-Polar Composition as aFunction of the Temperature

The viscosity of a host non-polar composition varies significantly withtemperature. If an associative polymer herein described is at aconcentration less than c* in toluene as the host non-polar composition,a particular deformation rate produces a lower stress if the flow occurswhen the liquid is at a higher temperature. A table indicating values ofviscosities for exemplary host composition liquids at a pressure of 1atm and at a temperature of 300 K is provided in FIG. 83.

For example, if the temperature is increased to 50° C. from 10° C., thehydrodynamic force imparted to the polymer at identical elongation ratesin the flow would decrease as the ratio of the viscosity at 50° C. tothe viscosity at 10° C.: (0.4400/0.6659)=0.661. That is, thehydrodynamic force for the same elongation rate, for the same polymer inthe same host non-polar composition would be on ⅔ as large at 50° C. asthe hydrodynamic force at 10° C.

Over the same temperature range, the density only changes by(0.87610−0.83870)/0.87610=4.3%.

Example 77: Density and Viscosity of a Non-Polar Composition as aFunction of the Temperature

Although the rupture force of a polymer is not proportional to theactivation bond enthalpy of the bond, the rank ordering of the ruptureforce can be inferred from large differences in the average bondenthalpies. For example, silicon-carbon single bonds have asubstantially lower average bond enthalpy than carbon-carbon singlebonds, given in FIG. 84.

Therefore, the selection of a polymer backbone that has exclusivelycarbon-carbon backbone bonds will enable associative polymer hereindescribed in a given host non-polar composition to move through a givenflow without breaking, even though that same flow might rupture abackbone that contains silicon-carbon bonds.

The rupture force for polymer chains that have exclusively C—C singlebonds in their backbone have F_(b) on average near 4 nN and backbonesthat contain Si—C single bonds in their backbone have F_(b) on averagenear 2 nN.

Consider one associative polymer, denoted (1), has repeat units thathave exclusively sp³ carbon in the backbone of the framing polymer, andanother associative polymer denoted (2), has approximately 10% Si—Cbonds in an otherwise sp³ carbon backbone. A pair of polymers isprepared such that their longest spans have nearly matched contourlength of their longest span, L=1000 nm. Consequently the two polymersalso have nearly matched c*. Associative non-polar compositions areprepared at ½c* in Linseed Oil and flow through a contraction thatimposes an elongation rate of 100,000 s⁻¹. The hydrodynamic force theyexperience are nearly matched: F_(K)=(1×10⁻⁶ m)(0.0331 N·s/m²)(10⁵s⁻¹)=3.3 nN, where the viscosity of Linseed oil is given in FIG. 82.

The hydrodynamic force exerted on the two polymers is the same, howeverpolymer (2) has a weaker backbone. F_(K) is greater than the ruptureforce of polymer (2), which is F_(b)=2 nN. The polymer degrade as theymove through the flow.

The same hydrodynamic force is less than the rupture force of polymer(1), which is F_(b)=4 nN. The polymers move through the flow without anyof their backbone bonds breaking.

Example 78: Determination of Rupture Longest Span forPolyethylhexylacrylate (PEHA)

In associative polymers herein described, the rupture contour lengthL_(b) indicates the shortest length of a longest span that, forspecified flow conditions and non-polar composition, will break (hereinalso indicated as rupture longest span). Thus, associative polymermolecules that have a contour length of their longest span equal to orgreater than the contour length L_(b) of the rupture longest span of theassociative polymer in the non-polar composition during the flow of itsintended application, those polymers would break and their benefit woulddecrease or be lost.

By design the distribution of longest span in associative polymers haveonly a small fraction of molecules that will degrade during use becausethe average longest span is less than the rupture longest span for theframing polymer. That is, only the high molecular weight end of thedistribution which contains individual molecules that will break,leaving a sufficient population of associative polymer intact tocontinue to deliver the desired rheological effect. A valuable productcan be obtained even if the distribution of polymers as synthesizedcontains some molecules that would break during use. The guidance hereinprovided relates unimodal distributions that contain a substantialfraction of polymers that do not break and will give sustainedbeneficial rheological effects.

An exemplary determination of the rupture longest span is hereinprovided with respect to polyethylhexylacrylate (PEHA) herein providedas an exemplary associative polymer.

A viscosity index improver is being developed for use in a synthetic oilthat has viscosity μ_(h)=155 mPa·s at 30° C. and υ_(h)=38 mPa·s at 60°C. is being developed using polyethylhexylacrylate (PEHA) as thebackbone of the framing polymer. To have lasting benefits, the longestspan will be kept shorter the contour length of the rupture longest spanof polyethylhexylacrylate when used in the synthetic oil at atemperature of 30° C. in a flow with a maximum velocity of 60 m/sthrough a 0.5 cm gap that is 1 m wide.

In preparation for synthesizing trial materials, the rupture longestspan of polyethylhexylacrylate is evaluated using a graphical method andconverted to the corresponding weight-average molecular weight of thelinear associative polyethylhexylacrylates that will be synthesized forfurther experimentation.

At 30° C., the viscosity the synthetic oil is h=0.155 μ_(a)·s and thedensity is ρ_(h)=842 kg/m³, giving a kinematic viscosity v_(h)=1.85×10⁻⁴m²/s.

The hydraulic diameter is calculated as μ_(h)=4 (cross sectionalarea)/(perimeter of the cross section)=4 (0.005 m) (1 m)/2(1.005 m)=0.01m

The Reynolds number of the flow is calculated using the host properties,the maximum velocity U=60 m/s and the hydraulic diameter of therectangular channel, d=0.01 m: Re=Ud/v_(h)=(60 m/s) (0.01 m)/(1.85×10⁻⁴m²/s)=(0.6)/(1.85)×10⁴.

To determine the value of the contour length of the rupture longest spanfor PEHA, a value of F_(b)=4 nN is selected as a good estimate of thestrength of the backbone because the PEHA backbone consists of sp³carbon-carbon bonds. Therefore, a graph as shown in FIG. 85 is madeshowing the increase of the hydrodynamic force F_(K) as a function of Lthe contour length of the longest span, with a horizontal line shown atthe value of F_(b). The intersection point was identified and its Lvalue was read from the graph.

The value of L at which F_(K) crosses the bond strength F_(b)=4 nN isL_(s)=930 nm.

The maximum number of backbone bonds in the longest span is calculatedfrom Ls using the average length of a C—C single bond, 0.154 nm, andusing the tetrahedral angle to compute the projection of the bond lengthalong the backbone as sin(109°/2)=0.818: n_(b)=L_(b)/(0.818*0.154)=7382backbone bonds

Each repeat unit of PEHA contributes n_(o)=2 backbone bond to the chain,so the rupture longest span corresponds to a degree of polymerizationDP_(b)=3691.

Each repeat unit of PEHA has a molar mass 184 g/mol, so the rupturelongest span corresponds to M_(wb)=6.79×10⁵ g/mol.

Regarding this number as approximate, a plan for experiments was madeusing three PEHA linear polymers with associative functional groups attheir end:

Short M_(w1)=3.40×10⁵ g/molMedium M_(w2)=4.10×10⁵ g/molLong M_(w3)=6.79×10⁵ g/mol

Telechelic PEHA of each length will be prepared using a polymerizationmethod that produces unimodal distributions with M_(w)/M_(n)<2. Theresearch program will proceed to evaluate the relative merits of thesethree molecular weights using ASTM and proprietary tests.

Example 79: Determination of Hydrodynamic Stability of Host Silicone OilComposition

To improve the performance of silicone-based heat transfer oils,polymers are being evaluated as drag reducing agents. The longestsilicone backbones available for this exemplary application are 400,000g/mol polydimethylsiloxane. To find out if there is any risk of chainscission, compare the longest available contour length to the contourlength of the rupture longest span if it were used in a silicone heattransfer oil of interest at the flow conditions in the heat transferequipment. In the case of polydimethyl siloxanes, the longest availablechains have (400,000 g/mol)/(74 g/mol)=5400 repeat units. Each repeatunit contributes two backbone bonds. The backbone bonds of siloxane are0.163 nm (longer than carbon-carbon bonds) and have a bond angle of 130°(a more open angle than sp³ carbon). Therefore, each monomer unitcontributes (0.163)(0.906)=0.148 nm to the backbone length. So a chainof the maximum molecular weight available has a contour length of:Longest available L=(5400)(2)(0.148 nm)=1597 nm.

Based on experience and the literature, siloxane backbone polymers havebackbones of similar strength to carbon-carbon backbones. The universalscaling relation shown in FIG. 86 for three different polymers, two thathave carbon-carbon backbones and one that has ether linkages. Based onthe similarity in backbone strength of siloxane backbones tocarbon-carbon backbone strength, the relationship is used to examine thefeasibility of using silicone polymers to create drag reducing agentsfor silicone heat transfer fluids.

The value of the group of coefficients shown as a function of Reynoldsnumber Re in FIG. 86 is the value at the rupture force; when used inplace of the corresponding group in the right hand side of the equationfor F_(K) shown in FIG. 86 the result is the rupture force for thepolymer, F_(b).

In the exemplary heat transfer system of interest, the velocities reach3 m/s in a tube with inner diameter 12 mm using a silicone oil as theheat transfer fluid. When used at or above 25° C. the viscosity neverexceeds 1.4 mPa·s. The density is 852 kg/m³ and its temperaturedependence can be neglected for initial evaluation purposes.

Thus, the Reynolds number of the flow is approximately: Re=(852kg/m³)*(3 m/s)*(0.012 m)/(0.0014 μ_(a)·s)=2.19×10⁴.

Using the graph the approximate value of the group of coefficients isapproximately 3×10⁻⁴ pN at the threshold for rupture for polymers thatthat have backbone strengths similar to the polymers being considered(polysiloxanes). Knowing the density and viscosity of the host siliconeoil composition and the diameter of tube through which it flows, anestimate of the length limitation on siloxane polymers:

Define the group of variables as A. Its value in N is A=3×10⁻¹⁶ N

Solving for L_(b) ²/ln(L_(b)/1 nm) gives:

L _(b) ²/ln(L _(b)/1 nm)=2.39×10¹¹ m²=2.39×10⁷ nm²

Check if the longest available chains would reach this limit using thecontour length of the longest available PDMS: (1597nm)²/ln(1597)=3.46×10⁵ nm².

The longest available silicone backbones will not break in the intendeduse. The project can continue without concern about hydrodynamicdegradation of the drag reducing silicone polymers.

Example 80: Polymers of Isocyanurate [Node] Having the Same Longest Span

In the case of branched architectures, the radius of gyration provides agood measure of the average longest span. (Exceptional synthetic effortis required to produce molecules in which the arms are highly crowded,for example near the core of a many arm star, so they are excluded frompractical consideration.) For example, if a mixture of polymers containspolymers with three arms and a distribution of lengths of the threearms, the shortest of the arms pervades the volume already pervaded bythe longest two arms.

The contribution of each molecule to the measured radius of gyrationexposes its longest span. Consider the two molecules shown in FIG. 87.Panel A shows a polymer with three arms and Panel B shows a linearpolymer. The synthetic method used to make the two different polymersprovide the same statistical distribution of the lengths of the arms.For molecules shown in Panel A, the degree of polymerization of the armsare denoted q for the longest, p for the intermediate and m for theshortest.

Therefore, the longest span, highlighted in bold, has p+q repeat units.In this example the repeat units are polyoctylacrylate, eachcontributing two backbone bonds. So the number of backbone bonds in thelongest span of the molecule in Panel A is 2(p+q). In panel B themolecule only has two FG-chain-units and overall has a linear structurewith a backbone that is the longest span emphasized in bold. If the twochains in a molecule of Panel B have corresponding degree ofpolymerization (p and q) to the longer two arms of a molecule of PanelB, the two chains have the same number of backbone bonds and the samecontour length of their longest span.

Example 81: Polymers of Trioxymethyl Ethane Node Having the Same ShearDegradation Properties

Two types of polystyrene framing polymers are prepared as shown in FIG.88. Panel A shows molecules that have 3 [FGa-chain-]- units attached toa [node]. Panel B shows molecules that have 2 [FGa-chain-]- unitsattached to a [node]. The molecules are associative framing polymers andcan be present in a mixture in an associative non-polar composition. Forexample, polystyrene is soluble in kerosene, diesel and gasoline. Suchframing polymers could be used to improve fire safety of fuels or toimprove engine performance or to reduce drag in refined productpipelines that deliver fuels from a major refinery to a distributiondepot. They can be used to provide a combination of these beneficialeffects that occur when an associative non-polar composition is in aflow.

The method used to synthesize the polymers ensures that the[FGa-chain-]- units have essentially the same statistical distribution.Therefore, the chains in the mixture have substantially the samestatistical distribution of longest span degree of polymerization (p+q).Accordingly, the two types of framing polymers have substantially thesame distribution in the number of backbone bonds in the longest spanand substantially the same average contour length of their longest span.

Therefore, the mixture of the two types of associative framing polymerswill have substantially the same shear degradation behavior in a flow.If the longest span is selected using the methods of present disclosure,a majority of the polymers will remain intact during use and continue toprovide the intended rheological properties.

Example 82: Exemplary Modification of Side Chain without AlteringDegradation Threshold

Associative framing polymers are suitable for modification to conferadditional beneficial effects. FIG. 89 shows in Panel A an[FG-chain-]-FG polymer that has m styrene repeat units in the chain.FIG. 89 shows in Panel B a chain derived from the chain in Panel A. Ofthe m styrene units, q has been converted to bromostyrene units inpreparation to attach functional groups that serve as anti-static agentswhen the polymer is used in a non-polar composition. The total backbonedegree of polymerization is not altered, so p+q=m. The overalldistribution of longest span in the polymer of Panel A is retained inthe polymers of Panel B. The functional groups may be added to thepolymers of Panel B without altering the longest span and thereforewithout altering the degradation thresholds calculated for the hostnon-polar composition of interest in the flow of interest.

Example 83: Polymers of Norbornene Derivatives Having the Same ShearDegradation Properties

Two types of polynorbornene framing polymers are prepared as shown inFIG. 90. Panel A shows molecules that the structure of a FGa-chain-FGastatistical co-polymer having p norbornene imide units with atriisopropylsilyl group and q norbornene diester units and acorresponding 5(p+q) total number of backbone atoms. Panel B shows thestructure of a FGa-chain-FGa statistical co-polymer having p norborneneimide units without a triisopropylsilyl group and q norbornene diesterunits and a corresponding 5(p+q) total number of backbone atoms. Themolecules are associative framing polymers and can be present in amixture in an associative non-polar composition. For example, hostnon-polar composition can be kerosene, diesel and gasoline. Such framingpolymers could be used to improve fire safety of fuels or to improveengine performance or to reduce drag in refined product pipelines thatdeliver fuels from a major refinery to a distribution depot. They can beused to provide a combination of these beneficial effects that occurwhen an associative non-polar composition is in a flow.

The method used to synthesize the polymers ensures that the[FGa-chain-]- units have essentially the same statistical distribution.Therefore, the chains in the mixture have substantially the samestatistical distribution of longest span degree of polymerization (p+q).Accordingly, the two types of framing polymers have substantially thesame distribution in the number of backbone bonds in the longest spanand substantially the same average contour length of their longest span.

Therefore, the mixture of the two types of associative framing polymersit is expected to have substantially the same shear degradation behaviorin a flow. If the longest span is selected using the methods of presentdisclosure, a majority of the polymers will remain intact during use andcontinue to provide the intended rheological properties.

Example 84: Exemplary Approach in Designing Associative Polymers toControl Rheological Properties

In applications that use polymers to provide a beneficial effect on therheological properties of a non-polar composition, it is desirable todesign the polymer such that it is not expected to break during flow.Depending upon the application, the polymer might pass through pumps,filters, contractions or expansions. These flows are particular likelyto cause degradation of polymers. When a polymer is being designed for aspecific application, the flow conditions of the application arespecified and the non-polar composition that is the liquid under flow isalso specified. Associative polymers use parameters of the specifiedflow conditions and specified non-polar composition to provide molecularstructures that resist degradation in flow.

In turbulent flow, the forces that produce chain scission are thoseexerted on polymers at a moment when they are stretched to a length thatis similar to the length of their longest span. When the molecularconformation is elongated, the forces exerted on it by the flowingliquid are similar to the forces that would be exerted on a slender rodof the same length and diameter as the elongated conformation of thepolymer. The length of an elongated polymer cannot exceed the length ofthe longest span of the molecule. In flows that are strong enough tobreak covalent bonds, the longest span provides a useful approximationfor the length of the extended conformations that are produced in thebursts of elongation that occur in turbulent flow. The greater thelongest span, the greater the hydrodynamic force on the long, slenderconformation of the polymer. Specifically, the tension is highest nearthe center of the elongated conformation. The magnitude of the tensionincreases as the square of the length of the long, slender elongatedconformation of the molecule. In addition, the magnitude of the tensileforce is proportional to the viscosity of the surrounding fluid andlocal rate of deformation. As skilled person would be familiar with therate of deformation, the strain rate, the symmetric part of the velocitygradient tensor and the elongation rate, all of which provide means toquantify the tendency of a flow to stretch and orient a polymer (ratherthan merely translating or rotating it). When the tension reaches thepoint that the polymer backbone will break, degradation occurs.

To minimize degradation, the longest span is kept below the length atwhich the flow conditions of the intended application in the non-polarcomposition of the application would cause a polymer to break. Thechemical structure of the backbone of the framing polymer determines theforce required to break the backbone. Once a skilled person hasidentified the flow of interest and the non-polar composition ofinterest, the present disclosure teaches them how to determine therupture longest span for a specified chemical structure of the backboneof the framing polymer.

In summary, described herein are associative polymers capable ofcontrolling a physical and/or chemical property of non-polarcompositions that can be used when the non-polar composition is in aflow, and related compositions, methods and systems. Associativepolymers herein described have a non-polar backbone with a longest spanhaving a molecular weight that remains substantially unchanged under theflow conditions and functional groups presented at ends of the non-polarbackbone, with a number of the functional groups presented at the endsof the non-polar backbone formed by associative functional groupscapable of undergoing an associative interaction with anotherassociative functional group with an association constant (k) such thatthe strength of each associative interaction is less than the strengthof a covalent bond between atoms and in particular less than thestrength of a covalent bond between backbone atoms.

The term “substantially unchanged” indicates a change in a detectedparameter that is within 5% of the parameter with respect to a referencemeasurement of the same parameter. For example, when referred to themolecular weight of a longest span under certain flow conditions, theterm substantially unchanged indicates a modification up to 5% in themolecular weight of the longest span measured after the application ofthe flow conditions with respect to the molecular weight of the longestspan measured before application of the flow conditions. In associativepolymers herein described, having a longest span which has a Mw that issubstantially unchagend under the flow conditions of an associativenon-polar composition, will also have a substantially unchanged radiusof gyration Rg under the flow conditions. Accordingly the Rg measuredafter application of the flow conditions will have an up to 5%difference with respect to the Rg of the associative polymer beforeapplication of the flow conditions.

The examples set forth above are provided to give those of ordinaryskill in the art a complete disclosure and description of how to makeand use the embodiments of the associative polymers, materials,compositions, systems and methods of the disclosure, and are notintended to limit the scope of what the inventors regard as theirdisclosure. All patents and publications mentioned in the specificationare indicative of the levels of skill of those skilled in the art towhich the disclosure pertains.

The entire disclosure of each document cited (including patents, patentapplications, journal articles, abstracts, laboratory manuals, books, orother disclosures) in the Background, Summary, Detailed Description, andExamples is hereby incorporated herein by reference. All referencescited in this disclosure are incorporated by reference to the sameextent as if each reference had been incorporated by reference in itsentirety individually. However, if any inconsistency arises between acited reference and the present disclosure, the present disclosure takesprecedence.

The terms and expressions which have been employed herein are used asterms of description and not of limitation, and there is no intention inthe use of such terms and expressions of excluding any equivalents ofthe features shown and described or portions thereof, but it isrecognized that various modifications are possible within the scope ofthe disclosure claimed Thus, it should be understood that although thedisclosure has been specifically disclosed by preferred embodiments,exemplary embodiments and optional features, modification and variationof the concepts herein disclosed can be resorted to by those skilled inthe art, and that such modifications and variations are considered to bewithin the scope of this disclosure as defined by the appended claims.

It is also to be understood that the terminology used herein is for thepurpose of describing particular embodiments only, and is not intendedto be limiting. As used in this specification and the appended claims,the singular forms “a,” “an,” and “the” include plural referents unlessthe content clearly dictates otherwise. The term “plurality” includestwo or more referents unless the content clearly dictates otherwise.Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which the disclosure pertains.

Unless otherwise indicated, the term “alkyl” as used herein refers to alinear, branched, or cyclic saturated hydrocarbon group typicallyalthough not necessarily containing 1 to about 15 carbon atoms, or 1 toabout 6 carbon atoms, such as methyl, ethyl, n-propyl, isopropyl,n-butyl, isobutyl, t-butyl, octyl, decyl, and the like, as well ascycloalkyl groups such as cyclopentyl, cyclohexyl and the like.Generally, although again not necessarily, alkyl groups herein contain 1to about 15 carbon atoms. The term “cycloalkyl” intends a cyclic alkylgroup, typically having 4 to 8, or 5 to 7, carbon atoms. The term“substituted alkyl” refers to alkyl substituted with one or moresubstituent groups, and the terms “heteroatom-containing alkyl” and“heteroalkyl” refer to alkyl in which at least one carbon atom isreplaced with a heteroatom. If not otherwise indicated, the terms“alkyl” and “lower alkyl” include linear, branched, cyclic,unsubstituted, substituted, and/or heteroatom-containing alkyl and loweralkyl, respectively.

Unless otherwise indicated, the term “hydrocarbyl” as used herein refersto any univalent radical, derived from a hydrocarbon, such as, forexample, methyl or phenyl. The term “hydrocarbylene” refers to divalentgroups formed by removing two hydrogen atoms from a hydrocarbon, thefree valencies of which may or may not be engaged in a double bond,typically but not necessarily containing 1 to 20 carbon atoms, inparticular 1 to 12 carbon atoms and more particularly 1 to 6 carbonatoms which includes but is not limited to linear cyclic, branched,saturated and unsaturated species, such as alkylene, alkenylenealkynylene and divalent aryl groups, e.g., 1,3-phenylene,—CH₂CH₂CH₂-propane-1,3-diyl, —CH₂-methylene, —CH═CH—CH═CH—. The term“hydrocarbyl” as used herein refers to univalent groups formed byremoving a hydrogen atom from a hydrocarbon, typically but notnecessarily containing 1 to 20 carbon atoms, in particular 1 to 12carbon atoms and more particularly 1 to 6 carbon atoms, including butnot limited to linear cyclic, branched, saturated and unsaturatedspecies, such as univalent alkyl, alkenyl, alkynyl and aryl groups e.g.ethyl and phenyl groups.

Unless otherwise indicated, the term “heteroatom-containing” as in a“heteroatom-containing alky group” refers to a alkyl group in which oneor more carbon atoms is replaced with an atom other than carbon, e.g.,nitrogen, oxygen, sulfur, phosphorus or silicon, typically nitrogen,oxygen or sulfur. Similarly, the term “heteroalkyl” refers to an alkylsubstituent that is heteroatom-containing, the term “heterocyclic”refers to a cyclic substituent that is heteroatom-containing, the terms“heteroaryl” and “heteroaromatic” respectively refer to “aryl” and“aromatic” substituents that are heteroatom-containing, and the like. Itshould be noted that a “heterocyclic” group or compound may or may notbe aromatic, and further that “heterocycles” may be monocyclic,bicyclic, or polycyclic as described above with respect to the term“aryl.” Examples of heteroalkyl groups include alkoxyaryl,alkylsulfanyl-substituted alkyl, N-alkylated amino alkyl, and the like.Examples of heteroaryl substituents include pyrrolyl, pyrrolidinyl,pyridinyl, quinolinyl, indolyl, pyrimidinyl, imidazolyl,1,2,4-triazolyl, tetrazolyl, and others known to a skilled person, andexamples of heteroatom-containing alicyclic groups are pyrrolidino,morpholino, piperazino, piperidino, and other known to a skilled person.

Unless otherwise indicated, the term “alkoxy” as used herein intends analkyl group bound through a single, terminal ether linkage; that is, an“alkoxy” group may be represented as —O-alkyl where alkyl is as definedabove. A “lower alkoxy” group intends an alkoxy group containing 1 to 6carbon atoms. Analogously, “alkenyloxy” and “lower alkenyloxy”respectively refer to an alkenyl and lower alkenyl group bound through asingle, terminal ether linkage, and “alkynyloxy” and “lower alkynyloxy”respectively refer to an alkynyl and lower alkynyl group bound through asingle, terminal ether linkage.

Unless otherwise indicated, the term “alkylamino” as used herein intendsan alkyl group bound through a single terminal amine linkage; that is,an “alkylamino” may be represented as —NH-alkyl where alkyl is asdefined above. A “lower alkylamino” intends an alkylamino groupcontaining 1 to 6 carbon atoms. The term “dialkylamino” as used hereinintends two identical or different bound through a common amine linkage;that is, a “dialkylamino” may be represented as —N(alkyl)₂ where alkylis as defined above. A “lower dialkylamino” intends an alkylaminowherein each alkyl group contains 1 to 6 carbon atoms. Analogously,“alkenylamino”, “lower alkenylamino”, “alkynylamino”, and “loweralkynylamino” respectively refer to an alkenyl, lower alkenyl, alkynyland lower alkynyl bound through a single terminal amine linkage; and“dialkenylamino”, “lower dialkenylamino”, “dialkynylamino”, “lowerdialkynylamino” respectively refer to two identical alkenyl, loweralkenyl, alkynyl and lower alkynyl bound through a common amine linkage.Similarly, “alkenylalkynylamino”, “alkenylalkylamino”, and“alkynylalkylamino” respectively refer to alkenyl and alkynyl, alkenyland alkyl, and alkynyl and alkyl groups bound through a common aminelinkage.

Unless otherwise indicated, the term “aryl” as used herein, and unlessotherwise specified, refers to an aromatic substituent containing asingle aromatic ring or multiple aromatic rings that are fused together,directly linked, or indirectly linked (such that the different aromaticrings are bound to a common group such as a methylene or ethylenemoiety). Aryl groups can contain 5 to 24 carbon atoms, or aryl groupscontain 5 to 14 carbon atoms. Exemplary aryl groups contain one aromaticring or two fused or linked aromatic rings, e.g., phenyl, naphthyl,biphenyl, diphenylether, diphenylamine, benzophenone, and the like.“Substituted aryl” refers to an aryl moiety substituted with one or moresubstituent groups, and the terms “heteroatom-containing aryl” and“heteroaryl” refer to aryl substituents in which at least one carbonatom is replaced with a heteroatom, as will be described in furtherdetail infra.

Unless otherwise indicated, the term “arene”, as used herein, refers toan aromatic ring or multiple aromatic rings that are fused together.Exemplary arenes include, for example, benzene, naphthalene, anthracene,and the like. The term “heteroarene”, as used herein, refers to an arenein which one or more of the carbon atoms has been replaced by aheteroatom (e.g. O, N, or S). Exemplary heteroarenes include, forexample, indole, benzimidazole, thiophene, benzthiazole, and the like.The terms “substituted arene” and “substituted heteroarene”, as usedherein, refer to arene and heteroarene molecules in which one or more ofthe carbons and/or heteroatoms are substituted with substituent groups.

Unless otherwise indicated, the terms “cyclic”, “cyclo-”, and “ring”refer to alicyclic or aromatic groups that may or may not be substitutedand/or heteroatom containing, and that may be monocyclic, bicyclic, orpolycyclic. The term “alicyclic” is used in the conventional sense torefer to an aliphatic cyclic moiety, as opposed to an aromatic cyclicmoiety, and may be monocyclic, bicyclic or polycyclic.

Unless otherwise indicated, the terms “halo”, “halogen”, and “halide”are used in the conventional sense to refer to a chloro, bromo, fluoroor iodo substituent or ligand.

Unless otherwise indicated, the term “substituted” as in “substitutedalkyl,” “substituted aryl,” and the like, is meant that in the, alkyl,aryl, or other moiety, at least one hydrogen atom bound to a carbon (orother) atom is replaced with one or more non-hydrogen substituents.

Examples of such substituents can include, without limitation:functional groups such as halo, hydroxyl, sulfhydryl, C1-C24 alkoxy,C2-C24 alkenyloxy, C2-C24 alkynyloxy, C5-C24 aryloxy, C6-C24 aralkyloxy,C6-C24 alkaryloxy, acyl (including C2-C24 alkylcarbonyl (—CO— alkyl) andC6-C24 arylcarbonyl (—CO-aryl)), acyloxy (—O-acyl, including C2-C24alkylcarbonyloxy (—O—CO-alkyl) and C6-C24 arylcarbonyloxy (—O—CO-aryl)),C2-C24 alkoxycarbonyl (—(CO)—O-alkyl), C6-C24 aryloxycarbonyl(—(CO)—O-aryl), halocarbonyl (—CO)—X where X is halo), C2-C24alkylcarbonato (—O—(CO)—O-alkyl), C6-C24 arylcarbonato (—O—(CO)—O-aryl),carboxy (—COOH), carboxylato (COO⁻), carbamoyl (—(CO)—NH2), mono-(C1-C24alkyl)-substituted carbamoyl (—(CO)—NH(C1-C24 alkyl)), di-(C1-C24alkyl)-substituted carbamoyl (—(CO)—N(C1-C24 alkyl)₂), mono-(C5-C24aryl)-substituted carbamoyl (—(CO)—NH-aryl), di-(C5-C24aryl)-substituted carbamoyl (—(CO)—N(C5-C24 aryl)2), di-N—(C1-C24alkyl), N—(C5-C24 aryl)-substituted carbamoyl, thiocarbamoyl(—(CS)—NH2), mono-(C1-C24 alkyl)-substituted thiocarbamoyl(—(CO)—NH(C1-C24 alkyl)), di-(C1-C24 alkyl)-substituted thiocarbamoyl(—(CO)—N(C1-C24 alkyl)₂), mono-(C5-C24 aryl)-substituted thiocarbamoyl(—(CO)—NH-aryl), di-(C5-C24 aryl)-substituted thiocarbamoyl(—(CO)—N(C5-C24 aryl)2), di-N—(C1-C24 alkyl), N—(C5-C24aryl)-substituted thiocarbamoyl, carbamido (—NH—(CO)—NH₂), cyano(-C≡N),cyanato (—O—C≡N), thiocyanato (—S—C≡N), formyl (—(CO)—H), thioformyl((CS)—H), amino (—NH2), mono-(C1-C24 alkyl)-substituted amino,di-(C1-C24 alkyl)-substituted amino, mono-(C5-C24 aryl)-substitutedamino, di-(C5-C24 aryl)-substituted amino, C2-C24 alkylamido(—NH—(CO)-alkyl), C6-C24 arylamido (—NH—(CO)-aryl), imino (—CR═NH whereR=hydrogen, C1-C24 alkyl, C5-C24 aryl, C6-C24 alkaryl, C6-C24 aralkyl,and others known to a skilled person), C2-C20 alkylimino (CR═N(alkyl),where R=hydrogen, C1-C24 alkyl, C5-C24 aryl, C6-C24 alkaryl, C6-C24aralkyl, and others known to a skilled person), arylimino (—CR═N(aryl),where R=hydrogen, C1-C20 alkyl, C5-C24 aryl, C6-C24 alkaryl, C6-C24aralkyl, and others known to a skilled person), nitro (—NO2), nitroso(—NO), sulfo (—SO2-OH), sulfonato (—SO2-O⁻), C1-C24 alkylsulfanyl(—S-alkyl; also termed “alkylthio”), C5-C24 arylsulfanyl (—S-aryl; alsotermed “arylthio”), C1-C24 alkylsulfinyl (—(SO)-alkyl), C5-C24arylsulfinyl (—(SO)-aryl), C1-C24 alkylsulfonyl (—SO₂-alkyl), C5-C24arylsulfonyl (—SO₂-aryl), boryl (—BH2), borono (—B(OH)₂), boronato(—B(OR)₂ where R is alkyl or other hydrocarbyl), phosphono (—P(O)(OH)₂),phosphonato (—P(O)(O⁻)₂), phosphinato (—P(O)(O⁻)), phospho (—PO₂),phosphino (—PH₂), silyl (—SiR₃ wherein R is hydrogen or hydrocarbyl),and silyloxy (—O-silyl); and the hydrocarbyl moieties C1-C24 alkyl (e.g.C1-C12 alkyl and C1-C6 alkyl), C2-C24 alkenyl (e.g. C2-C12 alkenyl andC2-C6 alkenyl), C2-C24 alkynyl (e.g. C2-C12 alkynyl and C2-C6 alkynyl),C5-C24 aryl (e.g. C5-C14 aryl), C6-C24 alkaryl (e.g. C6-C16 alkaryl),and C6-C24 aralkyl (e.g. C6-C16 aralkyl).

Unless otherwise indicated, the term “acyl” refers to substituentshaving the formula —(CO)-alkyl, —(CO)-aryl, or —(CO)-aralkyl, and theterm “acyloxy” refers to substituents having the formula —O(CO)-alkyl,—O(CO)-aryl, or —O(CO)-aralkyl, wherein “alkyl,” “aryl, and “aralkyl”are as defined above.

Unless otherwise indicated, the term “alkaryl” refers to an aryl groupwith an alkyl substituent, and the term “aralkyl” refers to an alkylgroup with an aryl substituent, wherein “aryl” and “alkyl” are asdefined above. In some embodiments, alkaryl and aralkyl groups contain 6to 24 carbon atoms, and particularly alkaryl and aralkyl groups contain6 to 16 carbon atoms. Alkaryl groups include, for example,p-methylphenyl, 2,4-dimethylphenyl, p-cyclohexylphenyl,2,7-dimethylnaphthyl, 7-cyclooctylnaphthyl,3-ethyl-cyclopenta-1,4-diene, and the like. Examples of aralkyl groupsinclude, without limitation, benzyl, 2-phenyl-ethyl, 3-phenyl-propyl,4-phenyl-butyl, 5-phenyl-pentyl, 4-phenylcyclohexyl, 4-benzylcyclohexyl,4-phenylcyclohexylmethyl, 4-benzylcyclohexylmethyl, and the like. Theterms “alkaryloxy” and “aralkyloxy” refer to substituents of the formula—OR wherein R is alkaryl or aralkyl, respectively, as just defined.

When a Markush group or other grouping is used herein, all individualmembers of the group and all combinations and possible subcombinationsof the group are intended to be individually included in the disclosure.Every combination of components or materials described or exemplifiedherein can be used to practice the disclosure, unless otherwise stated.One of ordinary skill in the art will appreciate that methods, deviceelements, and materials other than those specifically exemplified can beemployed in the practice of the disclosure without resort to undueexperimentation. All art-known functional equivalents, of any suchmethods, device elements, and materials are intended to be included inthis disclosure. Whenever a range is given in the specification, forexample, a temperature range, a frequency range, a time range, or acomposition range, all intermediate ranges and all subranges, as wellas, all individual values included in the ranges given are intended tobe included in the disclosure. Any one or more individual members of arange or group disclosed herein can be excluded from a claim of thisdisclosure. The disclosure illustratively described herein suitably canbe practiced in the absence of any element or elements, limitation orlimitations which is not specifically disclosed herein.

A number of embodiments of the disclosure have been described. Thespecific embodiments provided herein are examples of useful embodimentsof the disclosure and it will be apparent to one skilled in the art thatthe disclosure can be carried out using a large number of variations ofthe devices, device components, methods steps set forth in the presentdescription. As will be obvious to one of skill in the art, methods anddevices useful for the present methods can include a large number ofoptional composition and processing elements and steps.

In particular, it will be understood that various modifications may bemade without departing from the spirit and scope of the presentdisclosure. Accordingly, other embodiments are within the scope of thefollowing claims.

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1. A framing associative polymer comprising: a linear, branched, orhyperbranched polymer backbone having at least two ends and a cluster offunctional groups presented at two or more ends of the at least two endsof the linear, branched, or hyperbranched polymer backbone; wherein thelinear, branched, or hyperbranched polymer backbone is substantiallysoluble in a host non-polar composition, wherein a difference insolubility parameters between at least one framing associative polymerand the host composition gives rise to a Flory-Huggins interactionparameter (χ) of about 0.5 or less, wherein the cluster of thefunctional groups presented at the two or more ends is formed byassociative functional groups each capable of undergoing an associativeinteraction with another associative functional group in the non-polarcomposition with an association constant (k) wherein${k\left( M^{- 1} \right)} \geq {\frac{\frac{4}{3}{\pi\left( {R_{g}}^{2} \right)}^{\frac{3}{2}}N_{a}}{n_{F}} \times 10^{{- 2}3}}$in which R_(g) is the radius of gyration of the framing associativepolymer in the non-polar composition in nanometers, N_(a) is Avogadro'sconstant; and n_(F) is the average number of the associative functionalgroups in the framing associative polymer, wherein a longest span of theframing associative polymer has a contour length L_(f), such that½L_(bf)≤L_(f)<L_(bf), wherein L_(bf) is a rupture length of the framingassociative polymer in nanometers when the framing associative polymeris comprised within the host non-polar composition at framingassociative polymer concentration c to provide an associative non-polarcomposition in a flow, L_(bf) being given by implicit function$F_{bf} = {\frac{\pi\mu^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho\; d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³, wherein when c≤2c*, μ is the viscosity ofthe host non-polar composition μ_(h), ρ is the density of the hostnon-polar composition ρ_(h), and when c>2c*, μ is the viscosity of theassociative non-polar composition μ_(a), and ρ is the density of theassociative non-polar composition ρ_(a), and wherein${c^{*} = \frac{3M_{w}}{4{\left( {R_{g}}^{2} \right)}^{3/2}N_{a}}},$ inwhich M_(w) is the weight-average molecular weight, R_(g) is the radiusof gyration, and N_(a) is Avogadro's constant.
 2. (canceled) 3.(canceled)
 4. The framing associative polymer of claim 1, wherein theassociation constant is 6≤log₁₀ k≤14.
 5. (canceled)
 6. (canceled)
 7. Theframing associative polymer of claim 1, wherein the another associativefunctional group is presented at least one end of a differentassociative polymer.
 8. The framing associative polymer of claim 1,wherein the framing associative polymers are formed by at least twostructure units having formula [[FG-chain

-[node]_(z)

(I) and optionally one or more structural units having formula -

node

-

chain]

- (II), wherein: FG is an associative functional group (FGa); chain is anon-polar polymer substantially soluble in a non-polar composition, thepolymer having formula:R₁-[A]_(n)R₂  (III) wherein: A is a chemical and in particular anorganic moiety forming the monomer of the polymer; R₁ and R₂ areindependently selected from any carbon based or organic group with oneof R₁ and R₂ linked to an FG or a node and the other one of R₁ and R₂linked to an FG or a node; and n is an integer ≥1; z is 0 or 1; node isa covalently linked moiety linking one of R₁ and R₂ of at least onefirst chain with one of the R₁ and R₂ of at least one second chain;wherein the FG, chain and node of different structural units of thepolymer can be the same or different and wherein in at least onestructure unit having formula [[FG-chain

-[node]_(z)

(I) and optionally in one or more structural units having formula -

node

-

chain]

- (II), n is ≥250.
 9. The framing associative polymer of claim 8,wherein the associative functional group FGa is selected fromdiacetamidopyridine group, thymine group, Hamilton Receptor group,cyanuric acid group, carboxylic acid group, primary secondary ortertiary amine group, primary secondary and tertiary alcohol group, zincsulfonate, palladated sulfur-carbon-sulfur (SCS) pincer pyridine orprimary, secondary and tertiary amines.
 10. The framing associativepolymer of claim 8, wherein A is a diene, olefin, styrene,acrylonitrile, methacrylate or acrylate, vinyl acetate,dichlorodimethylsilane, tetrafluoroethylene, acids, esters, amides,amines, glycidyl ethers, or isocyanates, siloxane.
 11. The framingassociative polymer of claim 8, wherein n is equal to or greater than200 or equal to or greater than
 800. 12. An associative non-polarcomposition comprising: a host composition having a viscosity μ_(h), adensity ρ_(h), a dielectric constant equal to or less than about 5, andat least one framing associative polymer of claim 1, soluble in the hostcomposition and having a longest span with a countour length½L_(bf)≤L_(f)<L_(bf), wherein a difference in solubility parametersbetween the at least one framing associative Polymer and the hostcomposition gives rise to a Flory-Huggins interaction parameter (χ) ofabout 0.5 or less, wherein the at least one framing associative polymeris comprised in the host composition in a concentration c between fromabout 0.01 c*to about 10c*wherein${c^{*} = \frac{3M_{w}}{4{\left( {R_{g}}^{2} \right)}^{3/2}N_{a}}},$ inwhich M_(w) is the weight-average molecular weight, R_(g) is the radiusof gyration, and N_(a) is Avogadro's constant, wherein L_(bf) is arupture length of the framing associative polymer in nanometers when theframing associative polymer is within the host non-polar composition ata concentration c to provide the associative non-polar composition in aflow, L_(bf) being given by implicit function$F_{bf} = {\frac{\pi\mu^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho\; d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³, wherein, when c≤2c*, μ is μ_(h), and ρ isρ_(h), and when c>2c*, μ is the viscosity of the associative non-polarcomposition μ_(a), and ρ is the density of the associative non-polarcomposition ρ_(a), and wherein the concentration is suitable to controlat least one rheological property of the host composition.
 13. Thenon-polar composition of claim 12, wherein when the at least onerheological property is selected from drag reduction, mist control,lubrication and/or viscoelastic properties.
 14. (canceled)
 15. Thenon-polar composition of claim 12, wherein when the concentration of theat least one framing associative polymer is between 0.1 c* and c*. 16.The non-polar composition of claim 12, wherein the concentration of theat least one framing associative polymer is between 2c*to 10c*.
 17. Thenon-polar composition of claim 12, wherein the at least one framingassociative polymer comprises one or more pairs of telechelic polymers.18. The non-polar composition of claim 17, wherein the one or more pairsof telechelic polymers comprise complementary associative functionalgroups, wherein the complementary associative functional groups compriseone or more of carboxylic acids with other carboxylic acids, carboxylicacids with amines, alcohols with amines, alcohols with carboxylic acids,diacetamidopyridine with thymine, the Hamilton Receptor with cyanuricacid, zinc sulfonate or palladated sulfur-carbon-sulfur (SCS) pincerwith pyridine or primary, secondary or tertiary amines.
 19. Thenon-polar composition of claim 12, further comprising a cappingassociative polymer at a concentration up to 20 wt % of a totalassociative polymer concentration in the non-polar composition, whereinthe capping associative polymer comprises: a linear, branched, orhyperbranched polymer backbone having at least two ends and anassociative functional groups presented at one end of the at least twoends of the backbone, wherein the linear or branched backbone issubstantially soluble in the host composition, and the associativefunctional group is capable of undergoing an associative interactionwith another associative functional group in the non-polar compositionwith an association constant (k) wherein${k\left( M^{- 1} \right)} \geq {\frac{4}{3}{\pi\left( {R_{g}}^{2} \right)}^{\frac{3}{2}}N_{a} \times 10^{{- 2}3}}$in which Rg is the radius of gyration of the associative polymer in thenon-polar composition in nanometer, and N_(a) is Avogadro's constant.20. (canceled)
 21. (canceled)
 22. The non-polar composition of claim 12,wherein the host composition comprises crude oil, a refined fuel,lubricant, castor oil or a base oil.
 23. (canceled)
 24. A method tocontrol resistance to flow and/or flow rate enhancement of anassociative non-polar composition alone or in combination with anotherphysical and/or chemical property of the associative non-polarcomposition in a flow characterized by a Reynolds number Re, and acharacteristic length d, the method comprising providing a hostcomposition having a viscosity μ_(h), a density ρ_(h) and a dielectricconstant equal to or less than about 5; providing at least one framingassociative polymer of claim 1 substantially soluble in the hostcomposition and having a longest span with a contour length½L_(bf)≤L_(f)<L_(bf), the at least one framing associative polymerhaving a weight-average molecular weight equal to or higher to 200,000g/mol; and combining the host composition and the at least one framingassociative polymer at a concentration c selected between from about0.01 c* to 1c*, depending on the weight-average molecular weight and/orRadius of gyration of the at least one framing associative polymer andon a physical and/or chemical property to be controlled, wherein${c^{*} = \frac{3M_{w}}{4{\left( {R_{g}}^{2} \right)}^{3/2}N_{a}}},$ inwhich M_(w) is the weight-average molecular weight, R_(g) is the radiusof gyration, and N_(a) is Avogadro's constant wherein L_(bf) is arupture length of the at least one framing associative polymer innanometers when the at least one framing associative polymer is withinthe host non-polar composition at a concentration c to provide theassociative non-polar composition in a flow, L_(bf) being given byimplicit function$F_{bf} = {\frac{{{\pi\mu}_{h}}^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho_{h}d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$in which in which F_(bf) is the rupture force of the framing associativepolymer in nanonewtons, Re is the Reynolds number of the flow, d is thecharacteristic length of the flow in meters, μ_(h) is the viscosity ofthe host non-polar composition in Pa·s, and ρ_(h) is the density of thehost non-polar composition in kg/m³.
 25. The method of claim 24, whereinthe association constant of the at least one framing associative polymeris between 7≤log₁₀ k≤14.
 26. (canceled)
 27. (canceled)
 28. (canceled)29. The method of claim 24, wherein the concentration c is below orapproximately equal c* and the another physical and/or chemical propertyis enhanced lubrication.
 30. (canceled)
 31. (canceled)
 32. (canceled)33. (canceled)
 34. (canceled)
 35. The method of claim 30, wherein theconcentration c is below or approximately equal c* and the anotherphysical and/or chemical property is enhanced lubrication.
 36. A systemfor controlling a physical and/or chemical property of an associativenon-polar composition in a flow characterized by a Reynolds number Re,and a characteristic length d, the system comprising at least twobetween at least one framing associative polymer of claim 1 and at leastone host composition having a viscosity μ_(h), a density ρ_(h) and adielectric constant equal to or less than 5, wherein the at least oneframing associative polymer is substantially soluble in the hostcomposition, wherein a difference in solubility parameters between atleast one framing associative polymer and the host composition givesrise to a Flory-Huggins interaction parameter (χ) of about 0.5 or less,wherein the longest span of the framing associative polymer has acontour length ½L_(b)≤L_(f)<L_(b), wherein L_(b) is a rupture length ofthe framing associative polymer in nanometers when the framingassociative polymer is within the host non-polar composition in a flowto provide an associative non-polar composition wherein the framingassociative polymer is comprised at a concentration c, and L_(b) isgiven by the implicit function$F_{bf} = {\frac{\pi\mu^{2}{{Re}^{3/2}\left( L_{bf} \right)}^{2}}{4\rho\; d^{2}{\ln\left( L_{bf} \right)}} \times 10^{- 9}}$in which F_(bf) is the rupture force of the framing associative polymerin nanonewtons, Re is the Reynolds number, d is the characteristiclength of the flow in meters, μ is the viscosity of the host non-polarcomposition μ_(h) or the viscosity of the associative non polarcomposition μ_(a) in Pa·s, and ρ is the density of the host non-polarcomposition ρ_(h) or the viscosity of the associative non polarcomposition ρ_(a) in kg/m³, and wherein, when c≤2c*, μ is μ_(h), and ρis ρ_(h), and when c>2c*, μ is the viscosity of the associativenon-polar composition μ_(a), and ρ is the density of the associativenon-polar composition ρ_(a).
 37. The system of claim 36, furthercomprising further comprising a capping associative polymer at aconcentration up to 20 wt % of a total associative polymer concentrationin the non-polar composition, wherein the capping associative polymercomprises: a linear, branched, or hyperbranched polymer backbone havingat least two ends and an associative functional groups presented at oneend of the at least two ends of the backbone, wherein the linear orbranched backbone is substantially soluble in the host composition, andthe associative functional group is capable of undergoing an associativeinteraction with another associative functional group in the non-polarcomposition with an association constant (k) wherein${k\left( M^{- 1} \right)} \geq {\frac{4}{3}{\pi\left( {R_{g}}^{2} \right)}^{\frac{3}{2}}N_{a} \times 10^{{- 2}3}}$in which R_(g) is the radius of gyration of the associative polymer inthe non-polar composition in nanometer, and N_(a) is Avogadro'sconstant.
 38. The framing associative polymer of claim 1, wherein thecluster of functional groups comprises four or eight functional groups.39. The framing associative polymer of claim 1, wherein the cluster offunctional groups comprises a self-associative acid cluster.
 40. Theframing associative polymer of claim 1, having a weight-averagemolecular equal to or higher than about 60,000 g/mol.
 41. The framingassociative polymer of claim 1, wherein a functional group in thecluster of functional groups is capable of a self-to-self association.42. The framing associative polymer of claim 1, wherein the associativeinteraction is electrostatic interaction.
 43. The framing associativepolymer of claim 1, wherein the linear, branched, or hyperbranchedpolymer backbone comprises a poly(substituted alpha olefin),polyethylene or polypropylene substituted with alkoxycarbonyl.
 44. Theframing associative polymer of claim 1, wherein the linear, branched, orhyperbranched polymer backbone comprises polymethylacrylate orpolybutylacrylate.
 45. The framing associative polymer of claim 1,wherein the linear, branched, or hyperbranched polymer backbonecomprises polyacrylate with alkyl side chains.${k\left( M^{- 1} \right)} \geq {\frac{4}{3}{\pi\left( {R_{g}}^{2} \right)}^{\frac{3}{2}}N_{a} \times 10^{{- 2}3}}$